NON-DETERMINISTIC MATRICES: , THEORY AND ...

UNIVERSIDADE ESTADUAL DE CAMPINAS INSTITUTO DE FILOSOFIA E CIÊNCIAS HUMANAS

ANA CLAUDIA DE JESUS GOLZIO

NON-DETERMINISTIC MATRICES: THEORY AND APPLICATIONS TO ALGEBRAIC SEMANTICS MATRIZES NÃO-DETERMINÍSTICAS: TEORIA E APLICAÇÕES À SEMÂNTICA ALGÉBRICA

CAMPINAS 2017

Agência(s) de fomento e nº(s) de processo(s): FAPESP, 2013/04568-1

Ficha catalográfica Universidade Estadual de Campinas Biblioteca do Instituto de Filosofia e Ciências Humanas Cecília Maria Jorge Nicolau - CRB 8/3387

G584n

Golzio, Ana Claudia de Jesus, 1985GolNon-deterministic matrices : theory and applications to algebraic semantics / Ana Claudia de Jesus Golzio. – Campinas, SP : [s.n.], 2017. GolOrientador: Marcelo Esteban Coniglio. GolTese (doutorado) – Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas. Gol1. Lógica algébrica. 2. Lógica matemática não-clássica. 3. Categorias (Matemática). 4. Álgebra universal. I. Coniglio, Marcelo Esteban,1963-. II. Universidade Estadual de Campinas. Instituto de Filosofia e Ciências Humanas. III. Título.

Informações para Biblioteca Digital Título em outro idioma: Matrizes não-determinísticas : teoria e aplicações à semântica algébrica Palavras-chave em inglês: Algebraic logic Non-classical mathematical logic Categories (Mathematics) Universal algebra Área de concentração: Filosofia Titulação: Doutora em Filosofia Banca examinadora: Marcelo Esteban Coniglio [Orientador] Hércules de Araújo Feitosa Ciro Russo Hugo Luiz Mariano Newton Marques Peron Data de defesa: 30-03-2017 Programa de Pós-Graduação: Filosofia

UNIVERSIDADE ESTADUAL DE CAMPINAS INSTITUTO DE FILOSOFIA E CIÊNCIAS HUMANAS

A Comissão Julgadora dos trabalhos de Defesa de Tese de Doutorado, composta pelos Professores Doutores a seguir descritos, em sessão pública realizada em 30 de Março de 2017, considerou a candidata ANA CLAUDIA DE JESUS GOLZIO aprovada.

Prof. Dr. Marcelo Esteban Coniglio

Prof. Dr. Hércules de Araújo Feitosa

Prof. Dr. Ciro Russo

Prof. Dr. Hugo Luiz Mariano

Prof. Dr. Newton Marques Peron

A Ata de Defesa, assinada pelos membros da Comissão Examinadora, consta no processo de vida acadêmica da aluna.

To my parents: Altacir and João, To my husband: Claudecir, In memory of Mozart Luiz Carbonieri.

Acknowledgements I would like to express my sincere gratitude: To my parents Altacir and João and my husband Claudecir, for their support, encouragement and comprehension. To my supervisor Marcelo Esteban Coniglio, whom I admire, I respect, and I am immensely grateful for all his help and dedication during these years of work. To professors Itala M. L. D’Ottaviano and Walter A. Carnielli, that always offered good working conditions for them students. To professors Hércules de Araújo Feitosa and Luiz Henrique da C. Silvestrini, for the support, encouragement and friendship through my academic career so far. To professor Aldo Figallo-Orellano for his valuable contribution in this work. To professor Rodolfo Ertola Biraben for several suggestions. To longtime friends and study partners Kleidson Êglicio C. da S. Oliveira and Angela P. Rodrigues Moreira for the friendship and affection. To closer working friends: Thiago, Felipe, João, Alexandre and Edson for the relaxed companionship and for many suggestions. To other colleagues of Centre for Logic, Epistemology and History of Science (CLE) for the excellent companionship. To Mr. Travis D. Warwick (Head of the Kleene Mathematics Library at University of Wisconsin, in Madison, United States), and to professors Thomas Vougiouklis and Georges Hansoul by sending me bibliographical items that were fundamental to the writing of the first chapter. To FAPESP for the financial support (scholarship grant 2013/04568-1). To employees of CLE, for their good job that made this work possible. To everyone that in any way contributed to my personal and professional progress along these years.

“We hear within us the perpetual call: There is the problem. Seek its solution.” (David Hilbert)

Resumo Chamamos de multioperação qualquer operação que retorna, para cada argumento, um conjunto de valores ao invés de um único valor. Através das multioperações podemos definir uma estrutura algébrica munida com pelo menos uma multioperação. Esta estrutura é chamada de multiálgebra. O estudo delas começou em 1934, com a publicação de um artigo de Marty. No âmbito da Lógica, as multiálgebras foram consideradas por Avron e seus colaboradores sob o nome de matrizes não-determinísticas (ou Nmatrizes) e utilizadas como ferramenta semântica para a caracterização de algumas lógicas que não podem ser modeladas por uma única matriz finita. Carnielli e Coniglio introduziram a semântica de estruturas swap para LFIs (Lógicas da Inconsistência Formal), que são Nmatrizes definidas sobre ternas em uma álgebra booleana, que generaliza a semântica de Avron. Nesta Tese iremos apresentar um novo método de algebrização de lógicas, baseado em multiálgebras e em estruturas swap, que é similar ao método clássico de algebrização de Lindenbaum-Tarski, porém mais abrangente, porque podemos aplicá-lo a sistemas em que alguns operadores não são congruenciais. Em particular, este método será aplicado à uma família de lógicas modais não-normais e à algumas LFIs que não são algebrizáveis por nenhum método bem conhecido, incluindo a teoria geral de Blok e Pigozzi. Também obteremos teoremas de representação para alguns LFIs e provaremos que, sob a nossa abordagem, as classes de estruturas de swap para algumas extensões axiomáticas de mbC são subclasses da classe das estruturas swap para a lógica mbC. Palavras-chave: semântica algébrica; multiálgebra; matriz não-determinística; estrutura swap.

Abstract We call multioperation any operation that return for even argument a set of values instead of a single value. Through multioperations we can define an algebraic structure equipped with at least one multioperation. This kind of structure is called multialgebra. The study of them began in 1934 with the publication of a paper of Marty. In the realm of Logic, multialgebras were considered by Avron and his collaborators under the name of non-deterministic matrices (or Nmatrices) and used as semantics tool for characterizing some logics which cannot be characterized by a single finite matrix. Carnielli and Coniglio introduced the semantics of swap structures for LFIs (Logics of Formal Inconsistency), which are Nmatrices defined over triples in a Boolean algebra, generalizing Avron’s semantics. In this thesis we will introduce a new method of algebraization of logics based on multialgebras and swap structures that is similar to classical algebraization method of Lindenbaum-Tarski, but more extensive because it can be applied to systems such that some operators are non-congruential. In particular, this method will be applied to a family of non-normal modal logics and to some LFIs that are not algebraizable by method as Blok and Pigozzi general theory. We also will obtain representation theorems for some LFIs and we will prove that, under out approach, the classes of swap structures for some axiomatic extensions of mbC are a subclass of the class of swap structures for the logic mbC.

Keywords: algebraic semantics, multialgebra; non-deterministic matrix; swap structure.

Contents INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

0 0.1 0.2 0.3

BASIC CONCEPTS . . . . . . . . . . Notions of set theory . . . . . . . . . Notions of logic and universal algebra Notions of category theory . . . . . .

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1 1.1 1.2 1.3

ALGEBRAIC HYPERSTRUCTURES: Hypergroups . . . . . . . . . . . . . . Hyperlattices . . . . . . . . . . . . . . Multialgebras . . . . . . . . . . . . . .

ORIGINS . . . . . . . . . . . . . . . . . .

1.3.1 1.3.2

Homomorphisms of multialgebras Representation theorem . . . . . .

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Hyperrings and hyperfields. . . . . . Hv -structures . . . . . . . . . . . . . Non-deterministic matrices . . . . .

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1.4 1.5 1.6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and other concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2.1 2.2 2.3 2.4 2.5

SOME CONCEPTS IN UNIVERSAL MULTIALGEBRA . . . . . . . . . . Multialgebras and homomorphisms . . . . . . . . . . . . . . . . . . . . . . Submultialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpretation of formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multicongruences and quotient multialgebras . . . . . . . . . . . . . . . . The category of multialgebras . . . . . . . . . . . . . . . . . . . . . . . . .

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3 3.1 3.2

NON-DETERMINISTIC SEMANTICS FOR NON-NORMAL MODAL LOGICS . The Systems Tm, T4m, T45m, TBm and Dm . . . . . . . . . . . . . . . . . . . . . Swap structures for Tm, T4m, T45m, TBm and Dm . . . . . . . . . . . . . . . . .

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4 4.1 4.2 4.3

AN ALGEBRAIC STUDY OF LFIS BY MEANS OF Swap structures for CPL+ e . . . . . . . . . . . . . . . Swap structures for mbC . . . . . . . . . . . . . . . . Swap structures for some extensions of mbC . . . .

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SWAP STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 20 27

31 32 36

38 40 41 43 44

48 52 55 57 60

65 70

. 89 . 92 . 99 . 106

FINAL CONSIDERATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116

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Introduction We call algebraic hyperstructures the class of algebras in which the operations (called multioperations) return, for each argument, a set of values instead of a single value as in ordinary algebras. The study of hyperstructures began with the paper “Sur une généralisation de la notion de groupe” in 1934, by the French mathematician Frédéric Marty (MARTY, 1934). In that paper, the author introduces a definition of hypergroup, that is a generalization of usual notion of group by use of a multioperation. After Marty, several hyperstructures have emerged, including the multialgebras. A multialgebra (also known as hyperalgebra) is defined as a set provided with at least one multioperation. This concept was studied from many points of view and applied to several areas of Mathematics, Science Computer and Logic. In the realm of the Logic, multialgebras were used as semantis for logical systems. More recently, matrix semantics based on multialgebras were considered by Avron and his collaborators under the name of non-deterministic matrix (or Nmatrix) and used to characterize logics, in particular, some Logics of Formal Inconsistency - LFIs (see for instance (AVRON, 2005)). Semantics based on multialgebras for LFIs called swap structures were also proposed by Carnielli and Coniglio (CARNIELLI; CONIGLIO, 2016). The main motivation this Thesis is the study of multialgebras as a semantical tool for algebrization of logical systems and as an alternative algebraic structure (in the sense of the universal algebra). From the algebraic perspective, it is not so immediate that the generalization of basic concepts such as homomorphism, subalgebras and congruences works fine. Several different alternatives were proposed, but little was done with the purpose of being used in logic as semantics tool. So, the aim of this work is to present an algebraic study of multialgebras theory and to introduce a new algebraization method based on multialgebras and its properties, similar to the classical algebraization method of Lindenbaum-Tarski. Finally, we will apply this method to some LFIs and to a family of nonnormal modal systems that lie outside the scope of the usual techniques of algebraization of logics such as Blok and Pigozzi’s method. In order to achieve the objective proposed, this Thesis begins with a Chapter 0 where we will specify the notation and the basic concepts that will be used. We start showing basics notions of set theory and then we present some definitions, results and examples in the scope of universal algebra and logic. The reader who is already familiar with these concepts can pass quickly or even ignore the existence of this introductory chapter. To develop results in multialgebras, it was necessary to investigate what was

Introduction

12

already done in the literature. However, this task was not very easy due to the different classifications used to treat the same concepts. For instance, hyperlattice, that is a lattice with multioperations, was introduced by Benado with the name of multistructure (BENADO, 1953) and by Morgado (MORGADO, 1962) as reticuloide, however most of the authors, use the terms hyperlattice or multilattice. The names “multialgebra” and “hyperalgebra”, often are used with the same significance, but in 1950 Pickert called them “structures”. Already, a different approach, closer to relational systems, was given by Jonsson and Tarski that used the name of complex algebra (JONSSON; TARSKI, 1951; JONSSON; TARSKI, 1952). Many these authors have developed their concepts independently and so building a bridge between them is not a trivial task. The application of these structures to the context of the algebraic logic is still very little explored and authors such as Avron that uses these concepts in his definition of non-deterministic matrices, probably developed his work independently and so he does not refer to any of the works developed by authors such as Marty, Benado, Morgado, and so on. Since we consider important to establish a bridge between the logic and algebra in the framework of hyperstructures, then in Chapter 1 we will present a historical and analytical background of the main hyperstructures studied in literature until the case of the non-deterministic matrices used in logic. Our motivation to the study of multialgebras started before we knew the existing algebraic literature, even before we knew them with the name “multialgebras”. We only had information about the non-deterministic matrices developed by Avron and its collaborators. Initially, the idea of the project was to generalize concepts of universal algebra to nondeterministic matrices, because we believed in the possibility of obtaining advantages in terms of algebraization of logics. Thus we started our research with developments along this way, most of them are presented in Chapter 2. During the process of drafting that chapter, we discovered that the concept of multioperations used in the non-deterministic matrices was already applied in the literature of hyperstructures, and after an intensive investigation we found a broad literature about this subject. If, on the one hand, a large part of our original results was “lost” (in the sense that they were nothing new to compound a thesis), on the other hand we verified that we were on the correct way, since most of our concepts were in accordance with the published papers. Then, after we “walked over troubled water”, we started a new stage of the research with the goal of finding out what had already been done in this area (as much as possible). This research generated the Chapter 1. Of course, we do not change the initial goal of using multialgebras as a kind of algebraic semantic and in Chapters 3 and 4 we developed new algebraization tools using the multialgebras theory. Note that making developments in the framework of hyperstructures is not a simple task because there are different alternatives definitions of multialgebra, homomorphism, submultialgebra and so on. Thus, to make appropriate choices is not always obvious. In Chapter 2, we present which were our choices, by showing what are

Introduction

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the definitions of multialgebra, homomorphism between multialgebras, submultialgebra, homomorphic image of multialgebra, multicongruence and quotient multialgebra suitable for our purposes. In this chapter, we initiate a study of category for multialgebras which is used especially in Chapter 4. In particular it is shown that the category of multialgebras has arbitrary products. Note that other authors such as Nolan (NOLAN, 1979) has already done a categorial study of multialgebras, but our goal in this Thesis is not purely algebraic as in the works of Nolan. In Chapter 4, these results are applied to obtain representation theorems for some LFIs. In the future, we intend to do something similar with the family of non-normal modal systems studied in Chapter 3 of this Thesis. Chapter 3 is other important chapter of this Thesis. There we present some non-normal modal systems proposed by Ivlev in (IVLEV, 1988) in Hilbert-style deduction system. Later on algebraic semantics is proposed for these modal systems, consisting of swap structures (as introduced in (CARNIELLI; CONIGLIO, 2016) for LFIs), a kind of these swap structures constitute the original ϕ-valued non-deterministic matrices proposed by Avron for these systems, which were additionally studied by (CONIGLIO; CERRO; PERON, 2015). Then, we will present the so called Lindenbaum-Tarski swap structures for these logics and we will use a canonical valuation to show that the swap structures for these systems are correct and complete with respect to their respective Hilbert-style versions. This method is more simpler and natural than the proposed by Blok-Pigozzi and it is similar to the classical algebraization method of Lindenbaum-Tarski with the peculiarity that the use of multialgebras eliminates the requirement that all the operators be congruential. The Chapter 4 was developed with the collaboration of the Professor Aldo Figallo Orellano (Universidad Nacional del Sur, Bahía Blanca, Argentina) who is currently developing a postdoctoral at the CLE/Unicamp1 . Part of the results presented in that chapter are based on the pre-print (CONIGLIO; ORELLANO; GOLZIO, 2016). We started studying the (purely linguistic) expansion of CPL+ , denoted by CPL+ e . This expansion is nothing more than CPL+ defined by adding ¬ and ◦ to the language of CPL+ without any axioms or rules for them. In order to prove completeness, we will apply, once again the method of the Lindenbaum-Tarski swap structure already used in the Chapter 3 for modal systems. After this step, we will concentrate our efforts on the algebraic theory of KmbC , the class of swap structures for the logic mbC, the weakest system in the hierarchy of LFIs and on some extensions of mbC. Algebraic semantic of swap structures for these systems will be reintroduced in a slightly more general form, in order to define a hierarchy of classes of multialgebras associated to the corresponding hierarchy of logics. This is in line with the traditional approach of algebraic logic, in which hierarchies of classes of 1

The Centre for Logic, Epistemology and the History of Science (CLE) at the State University of Campinas (UNICAMP) is the place were this Thesis was developed.

Introduction

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algebraic models are associated to hierarchies of logics. We will show that the class KmbC is closed under sub-swap-structures and products, but it is not closed under homomorphic images, hence it is not a variety in the usual sense. And using the results of Chapter 2, we will present Birkhoff-style representation theorems for the class of each one of these logical systems. At last in the Final Considerations, we highlight the most important results of this Thesis and discuss the possible future works in the interaction between multialgebras and logics.

15

0 Basic concepts

This chapter is the theoretical background for the remainder of the work. The concepts will be presented briefly and they will be used as a way to standardize the notation.

0.1 Notions of set theory We assume that the reader is familiar with the basics notions of set theory. If it is not the case, we suggest specific books of set theory, as (HRBACEK; JECH, 1999; FEITOSA; NASCIMENTO; ALFONSO, 2011; PRISCO, 1997). We adopt the intuitive notion that a set is a collection of objects and call these objects elements of the set. Let A be a set. If a is an element of A, we write a ∈ A and, otherwise, a 6∈ A. Also, we assume that the basic operations in set theory have the usual meaning, they are: inclusion, union, intersection and difference, they are denoted by ⊆, ∪, ∩ and −, respectively. We use the symbols “=” to denote the equality and “6=” to denote the fact that two sets are not equal. If A and B are sets, then we define the operation of proper subset, in symbols A ⊂ B, meaning A ⊆ B and A 6= B. When A is a subset of a set B, we abbreviate B − A by A0 . To denote the empty set, that is, a set which has no elements, we use the symbol ∅. We say that two sets A and B are disjoints, when A ∩ B = ∅. We will use P(A) to denote the set of all subsets of a set A and we use the expression P(A)+ to denote the set of non-empty subsets of a set A, that is P(A) − {∅}. Definition 0.1.1 (Cartesian product). Let A and B be two sets. The cartesian product of A and B is the set: A × B = {(a, b) : a ∈ A and b ∈ B}, such that (a, b) are ordered pairs. If A and B are sets, then a binary relation (or just a relation on A × B) is any subset of A × B.

Chapter 0. Basic concepts

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Definition 0.1.2 (Finite cartesian product). Let n be a positive integer and A1 , A2 , . . . , An be sets. The Cartesian product A1 × A2 × . . . × An , of A1 , A2 , . . . , An is defined by: A1 × A2 × . . . × An =

n Y

Ai = {(a1 , a2 , . . . , an ) : a1 ∈ A1 , a2 ∈ A2 , . . . , an ∈ An },

1

such that (a1 , a2 , . . . , an ) are ordered n-tuples. If, in the above definition, A1 = A2 = . . . = An , we represent A1 × A2 × . . . × An by A and take A0 as {∅}. n

Let A be a set and n a positive integer. An n-ary relation R on A is defined as a subset of An and R is called a relation of type n. If an n-tuple is an element of an n-ary relation R, we denote this by (a1 , a2 , . . . , an ) ∈ R. In the case that S is a binary relation we use the notations (a1 , a2 ) ∈ S or a1 Sa2 with the same meaning. We can eventually write ~a to denote the n-tuple (a1 , . . . , an ) of elements in An . Definition 0.1.3 (Inverse relation, image and inverse image). If S is a binary relation on A × B, then (i) The inverse relation of S is a relation: S −1 = {(b, a) ∈ B × A : (a, b) ∈ S}. (ii) The image of C ⊆ A by S is the set: S[C] = {b ∈ B : ∃a ∈ C, such that (a, b) ∈ S}. In particular, the image of S is im(S) = S[A]. (iii) The inverse image of D ⊆ B by S is the set: S −1 [D] = {a ∈ A : ∃b ∈ D, and (a, b) ∈ S}. In particular, the domain of S is dom(S) = S −1 [B]. Definition 0.1.4 (Composition of relations). If S is a binary relation on A × B and R is a binary relation on B × C, then the composition of S and R is the relation: R ◦ S = {(a, c) ∈ A × C : ∃b ∈ B, such that (a, b) ∈ S and (b, c) ∈ R}. Definition 0.1.5 (Relation properties). If S is a relation on A, then we say that S is


17

(i) reflexive when, for all a ∈ A, we have aSa; (ii) symmetric when, for all a, b ∈ A, if aSb, then bSa; (iii) transitive when, for all a, b, c ∈ A, if aSb and bSc, then aSc; (iv) antisymmetric when, for all a, b ∈ A, if aSb and bSa, then a = b; Definition 0.1.6 (Equivalence relation). Let A be a set. We say that S is an equivalence relation on A, if S is a reflexive, symmetric and transitive relation. Definition 0.1.7 (Equivalence class). Let A be a set and S be an equivalence relation on A and a ∈ A. The equivalence class of a w.r.t.1 S is the set: [a]S = {b ∈ A : bSa}. Notation 0.1.8. When the equivalence relation is clear from the context, we can denote the class above only by [a]. Definition 0.1.9 (Quotient set). Let A be a set and S be an equivalence relation on A. The quotient set of A by the relation S, denoted by A/S, is the set of the equivalence classes of S, that is: A/S = {[a] : a ∈ A}. Definition 0.1.10 (Ordering relation). Let A a set. We say that S is an ordering relation (or partial ordering relation) on A, if S is a reflexive, antisymmetric and transitive relation. Definition 0.1.11 (Poset). A partially ordered set (or poset 2 ) is a pair hA, Si such that A is a nonempty set and S is a partial ordering relation on A. Definition 0.1.12. If hA, ≤i is a poset and B ⊆ A, then we say that: (i) an element a1 of A is an upper bound of B, when for all b ∈ B, we have that b ≤ a1 ; (ii) an element a2 of A é a lower bound of B, when for all b ∈ B, we have that a2 ≤ b; (iii) an element b1 of B is a maximum of B, when for all b ∈ B, we have that b ≤ b1 ; (iv) an element b2 of B is a minimum of B, when for all b ∈ B, we have that b2 ≤ b; (v) an element b1 of B is a maximal of B, when for all b ∈ B if, b1 ≤ b, then b = b1 ; (vi) an element b2 of B is a minimal of B, when for all b ∈ B if, b ≤ b2 , then b2 = b. 1 2

Abbreviation for “with respect to”. That is the usual abbreviation for partially ordered set.


18

Definition 0.1.13 (Supremum and infimum). Let hA, ≤i be a poset and B ⊆ A. The supremum of B (denoted by supB), if it exists, is the minimum of the set of all upper bounds of B and the infimum of B (denoted by inf B), if it exists, is the maximum of the set of all lower bounds of B. Definition 0.1.14 (Quasi ordering relation). Let A a set. We say that S is a quasi ordering relation on A, if S is a reflexive and transitive relation. Definition 0.1.15 (Chain). A chain C in a poset A = hA, ≤i is a subset of A such that, for all a, b ∈ C, a ≤ b or b ≤ a. Definition 0.1.16 (Function). Let A and B be sets. A function f from A to B, denoted by f : A → B is an binary relation on A × B, such that for all a ∈ A, there is exactly one b ∈ B with (a, b) ∈ f . In this case, we write f (a) = b. In a function f : A → B, the sets A and B are called of domain and range of f , respectively. The set of all functions from A to B will be denoted by B A . If A = 6 ∅, then ∅A = ∅. But, if A = ∅, then B ∅ = {∅}, because there is a single function ∅ : ∅ → B with empty domain and, in particular, ∅∅ = {∅}. Definition 0.1.17 (Image and inverse image). Let f : A → B a function. If C ⊆ A and D ⊆ B, then the sets f [C] = {f (c) : c ∈ C} and f −1 [D] = {a ∈ A : f (a) ∈ D} represent the image of C by the function f and the inverse image of D by the function f , respectively. Remark 0.1.18. If f : A → B is a function, ~b = (b1 . . . , bn ) ∈ B n (for n > 0) then f −1 (~b) will stand for {~a ∈ An : (f (a1 ), . . . , f (an )) = ~b}. Definition 0.1.19 (Constant function). A function f : A → B is called a constant function if, f [A] is a singleton. Definition 0.1.20 (Identity function). A function f : A → A is called the identity function if, f (a) = a for all a ∈ A. As well known, the composition of functions is still a function: Proposition 0.1.21 (Composition function). If f : A → B and g : B → C are two functions, then g ◦ f : A → C is a function, such that (g ◦ f )(a) = g(f (a)), for all a ∈ A. Definition 0.1.22 (Injection, surjection and bijection). A function f : A → B is called: (i) an injection (or one-to-one) when, for all a1 , a2 ∈ A, if f (a1 ) = f (a2 ), then a1 = a2 .


19

(ii) a surjection (or onto) when, for all b ∈ B, there is a ∈ A, such that f (a) = b. (iii) a bijection when it is an injection and a surjection. Definition 0.1.23 (Projection). Let A1 and A2 be sets, such that, a1 ∈ A1 and a2 ∈ A2 . The function πi : A1 × A2 → Ai defined by πi ((a1 , a2 )) = ai , is called projection on the ith coordinate of A1 × A2 , for i = 1, 2. Definition 0.1.24 (Indexed family of sets). Let I and A be sets. An indexed family of sets I, denoted by {fi }i∈I , is a function f : I → A. If i ∈ I, then we can denote f (i) by fi . The set I is called of index set of family {fi }i∈I . Definition 0.1.25 (Generalized union and intersection). If {Ai }i∈I family of subsets of a certain set A, then the union and the intersection of these sets are denoted, respectively, by: [ Ai = {a : a ∈ Ai for some i ∈ I} and i∈I

\

Ai = {a : a ∈ Ai for all i ∈ I 6= ∅}.

i∈I

Definition 0.1.26 (Infinite cartesian product). Let I be an index set and {Ai }i∈I be a family of sets indexed by I. The cartesian product of the sets in A is defined by: Y i∈I

Ai = {f : I →

[

Ai : f (i) ∈ Ai for all i}.

i∈I

Definition 0.1.27 (General projections). Let I be an index set, such that i, j ∈ I and Q {Ai }i∈I be a family of sets indexed by I. The function πj : i∈I Ai → Aj , defined by Q πj (f ) = f (j), is called of projection on the ith coordinate of i∈I Ai . Axiom of choice: For any set A there is a function f : P(A) → A, such that if ∅= 6 B ∈ P(A), then f (B) ∈ B. This function is called of choice function. In relation to the sets of numbers, we will denote the usual sets of natural, integers, rational, irrational and real numbers by N, Z, Q, I and R, respectively. We will present in the following some more specific properties of sets in general. Definition 0.1.28 (Filter). Let S be a set. A filter on S is a collection F of subsets of S that satisfies the following three conditions: (i) S ∈ F ; (ii) If A ∈ F and A ⊆ B ⊆ S, then B ∈ F ; (iii) If A ∈ F and B ∈ F , then A ∩ B ∈ F .


20

Definition 0.1.29 (Generated filter). Let X be a subset of S. A generated filter by X, on S is F = {A ⊆ S : X ⊆ A}. Definition 0.1.30 (Proper filter). A proper filter on S is a filter F , such that F 6= P(S). Definition 0.1.31 (Ultrafilter). Let S a set. A filter U on S is an ultrafilter on S if U is a proper filter and for all set A ⊆ S, we have that or A ∈ U or (S − A) ∈ U .

0.2 Notions of logic and universal algebra We will introduce the basic notions of logic in general, as well as some definitions and properties of the propositional calculus. For the reader who wants to know more about these topics, we suggest the books (SHOENFIELD, 1967; MENDELSON, 1987; FEITOSA; PAULOVICH, 2005). Definition 0.2.1 (Variables). The set of variables, denoted by Ξ, is a countable set of symbols defined by: Ξ = {ξn : n ∈ N}. Notation 0.2.2. We can denote a variable in Ξ by p or q with or without subscripts. Definition 0.2.3 (Signature). Let Ξ be a set of variables. A signature Θ is a family {Θn }n∈N , such that for all Θn , Ξ ∩ Θn = ∅ and if n 6= m, then Θn ∩ Θm = ∅. Elements of Θn are called operator symbols of arity n. Elements of Θ0 are called constants. The support of Θ is the set: |Θ| =

[

Θn = {c : c ∈ Θn for some n ∈ N}.

When there is no risk of confusion, we write Θ instead of |Θ|. Definition 0.2.4 (Propositional language). A propositional language with signature Θ on Ξ, denoted by L(Θ, Ξ), is the algebra defined by: (i) If ξ ∈ Ξ, then ξ ∈ L(Θ, Ξ); (ii) If c ∈ Θn and {α1 , α2 , . . . , αn } ∈ L(Θ, Ξ), then c(α1 , α2 , . . . , αn ) ⊆ L(Θ, Ξ); (iii) The set L(Θ, Ξ) is generated only by items (i) and (ii). If L(Θ, Ξ) satisfies the three conditions above, we say that the propositional language L(Θ, Ξ) is the algebra freely generated by the set Θ on Ξ.


21

Definition 0.2.5 (Formulas). If Θ is a signature and L(Θ, Ξ) is a propositional language over Θ, then the elements of L(Θ, Ξ) are called formulas over Θ. We can use the notation LΘ or F or(Θ) instead of L(Θ, Ξ) to denote the set of the formulas over Θ, omitting the set of variables Ξ, whenever it is clear in the context. We will use Greek lowercase letters, such as ϕ, ψ, α, β and γ, with or without index, to denote arbitrary formulas in L(Θ, Ξ). The set of variables that occurs in a formula α of L(Θ, Ξ) will be denoted by V ar(α). Definition 0.2.6. Let Θ be a signature and Ξ0 ⊂ Ξ. We denote by L(Θ, Ξ0 ) the subset of L(Θ, Ξ) formed by the formulas α, such that V ar(α) ⊆ Ξ0 . In particular, if Ξn = {ξi : 0 ≤ i ≤ n}, for n ≥ 0, then L(Θ, Ξn ) is the subset of L(Θ, Ξ) formed by formula schemes α, such that V ar(α) ⊆ {ξ0 , . . . , ξn }. Definition 0.2.7 (Consequence relation). Let Θ be a signature, L(Θ, Ξ) be a propositional language over Θ and Γ, ∆ ⊆ P(L(Θ, Ξ)). A consequence relation in L(Θ, Ξ) is a relation ` ⊆ P(L(Θ, Ξ)) × L(Θ, Ξ) that satisfies the following conditions: (i) If α ∈ Γ, then Γ ` α (strong reflexivity); (ii) If Γ ` α and Γ ⊆ ∆, then ∆ ` α (monotonicity); (iii) If Γ ` α and ∆ ` β, for all β ∈ Γ, then ∆ ` α (transitivity). Definition 0.2.8 (Propositional logic). A propositional logic is a pair L = hΘ, ì, such that Θ is a signature and ` is a consequence relation in L(Θ, Ξ). Remark 0.2.9. The classical propositional calculus (or classical propositional logic) is a particular case of propositional logic. We use CPC as abbreviation for Classical Propositional Calculus or Classical Propositional Logic and in a deduction we will use this abbreviation whenever we are referring to a result that hold in the CPC. Definition 0.2.10 (Theory). Let L = hΘ, ì be a propositional logic. A theory in L is a set T ⊆ L(Θ, Ξ) such that if T ` α, then α ∈ T . We denote the set of all the theories in L by T h(L). Definition 0.2.11 (Algebra). Let Θ be a signature. An algebra A over Θ (or of type Θ) is a pair hA, σA i such that A is a non-empty set, the universe (or support) of A, and σA is a function that assigns, for each n ∈ N and c ∈ Θn , an n-ary operation in A, cA : An → A.


22

In the sequel, sometimes we will refer to an algebra A = hA, σA i by means of its support A and the support of A will be frequently denoted by |A|. When there is no risk of confusion, we write σ instead of σA . Definition 0.2.12 (Similar algebras). Two algebras A = hA, σA i and B = hB, σB i are similar if they share the same type Θ Definition 0.2.13 (Logic and algebra of same type). If L = hΘ, ì is a propositional logic over Θ and A = hA, σi is an algebra, also over Θ, then we say the logic L and the algebra A have the same type. Definition 0.2.14 (Multifunction). Let A and B be nonempty sets. A multifunction (or multioperation) g from B to A, denoted by g : B →M A, is a function g : B → (P(A)−{∅}). Definition 0.2.15 (Partial multifunction). Let A and B be nonempty sets. A function g : B → P(A) is a partial multifunction g from B to A. Definition 0.2.16 (Multifunction composition). Let A, B and C nonempty sets, and g1 : A →M B, g2 : B →M C multifunctions. The multifunction composition is a multifunction S g2 ◦g1 : A →M C, given by (g2 ◦g1 )(a) = {g2 (b) : b ∈ g1 (a)}, for every a ∈ A. Proposition 0.2.17. The composition operation between multifunctions is associative, i.e. If g1 : A →M B, g2 : B →M C and g3 : C →M D are multifunctions, then (g3 ◦g2 )◦g1 = g3 ◦(g2 ◦g1 ). Proof. We must show that, for all a ∈ A, ((g3 ◦g2 )◦g1 )(a) = (g3 ◦(g2 ◦g1 ))(a). First of all: S given a ∈ A, (g2 ◦g1 )(a) = {g2 (b) : b ∈ g1 (a)}, while (g3 ◦(g2 ◦g1 ))(a) =

[

{g3 (c) : c ∈ (g2 ◦g1 )(a)}.

So, d ∈ (g3 ◦(g2 ◦g1 ))(a) if, and only if there is c ∈ (g2 ◦g1 )(a), such that d ∈ g3 (c) if, and only if there is b ∈ g1 (a) and there is c ∈ g2 (b) such that d ∈ g3 (c). (∗) On the other hand, if b ∈ B, (g3 ◦g2 )(b) = {g3 (c) : c ∈ g2 (b)}, while S

((g3 ◦g2 )◦g1 )(a) =

[

{(g3 ◦g2 )(b) : b ∈ g1 (a)}.

So, d ∈ ((g3 ◦g2 )◦g1 )(a) if, and only if there is b ∈ g1 (a), such that d ∈ (g3 ◦g2 )(b), if, and only if there is b ∈ g1 (a) and there is c ∈ g2 (b), such that, d ∈ g3 (c) i. e. if and only if the condition (∗) is true. This shows that the sets ((g3 ◦g2 )◦g1 )(a) and (g3 ◦(g2 ◦g1 ))(a) are the same, for all a ∈ A. Therefore, (g3 ◦g2 )◦g1 = g3 ◦(g2 ◦g1 ).


23

Definition 0.2.18 (Homomorphism). Let A = hA, σi and B = hB, σ 0 i be two similar algebras over Θ. A homomorphism h : A → B from A to B over Θ is a function h : A → B such that for all n ≥ 0, c ∈ Θn and a1 , . . . , an ∈ A, h(cA (a1 , . . . , an )) = cB (h(a1 ), . . . , h(an )). In particular, h(cA ) = cB , if c ∈ Θ0 . Definition 0.2.19 (Valuation). Let A = hA, σi an algebra over Θ. The homomorphisms h : L(Θ, Ξ) → A are called of valuations of L(Θ, Ξ) in A. Definition 0.2.20 (Subuniverse). Let A = hA, σi be an algebra over Θ. A subuniverse of A over Θ is a non-empty subset B of A which is closed under the operations of A, i. e., for all n ≥ 0, if c ∈ Θn and b1 , . . . , bn ∈ B, then cA (b1 , . . . , bn ) ∈ B. Definition 0.2.21 (Subalgebra). If A = hA, σi and B = hB, σ 0 i are two similar algebras over Θ, such that B is a subuniverse of A, then we say that B is a subalgebra of A over Θ and we denote this by B ⊆ A. Definition 0.2.22 (Subuniverse generated). Let A = hA, σi be an algebra over Θ and X ⊆ A. The subuniverse of A over Θ generated by X, denoted by sg(X), is defined as follows: \ sg(X) = {B : B is subuniverse of A over Θ and X ⊆ B}. Remark 0.2.23. Note that A is a subuniverse of A that contains X. So, the set {B : B is subuniverse of A over Θ and X ⊆ B} is non-empty and therefore sg(X) is well defined. Definition 0.2.24 (Subalgebra generated). Let hA, σi an algebra and X ⊆ A. We say that hA, σi is generated by X, if sg(X) = A. Definition 0.2.25 (Congruence). Let A = hA, σi be an algebra over a signature Θ and θ ⊆ A × A. Then θ is a congruence in A if: 1. θ is an equivalence relation; 2. for all n > 0, c ∈ Θn and a1 , ..., an , b1 , ..., bn ∈ A, if ai θbi , for all 1 ≤ i ≤ n, then cA (a1 , . . . , an )θcA (b1 , . . . , bn ). Remark 0.2.26. In the last definition (and throughout the text) we use the symbols Θ and θ to denote distinct things: Θ is used denoting signatures and θ denoting congruence or equivalence relations.


24

Definition 0.2.27 (Quotient algebra). If A = hA, σi is an algebra over a signature Θ and θ ⊆ A × A is a congruence in A, then the quotient algebra of A by θ, denoted by A/θ, is the algebra over Θ with support A/θ with the operations cA/θ (a1 /θ, . . . , an /θ) = cA (a1 , . . . , an )/θ, such that a1 , . . . , an ∈ A, n ≥ 0 and c ∈ Θn . In particular, cA/θ = cA /θ if c ∈ Θ0 . The operations in A/θ are well defined, because θ is a congruence in A. In what follows, we will present some examples of well known algebras in the literature, since they will be used throughout the thesis. Example 0.2.28 (Semigroup). A semigroup is an algebra hG, ·i such that · is a binary operation that satisfies the following identity: (G1) For all a, b, c ∈ G, a · (b · c) = (a · b) · c [associativity]; Example 0.2.29 (Group). A group is an algebra hG, ·, −1 , 1i such that the operations ·, −1 and 1 are, binary, unary and nullary, respectively and the following identities are satisfied: (G1) For all a, b, c ∈ G, a · (b · c) = (a · b) · c [associativity]; (G2) For all a ∈ G, a · 1 = a = 1 · a [identity element]; (G3) For all a ∈ G, a · a−1 = 1 = a−1 · a [inverse element]. Example 0.2.30 (Abelian group). A group hG, ·, −1 , 1i is abelian (or commutative) if the following identity is satisfied: (G4) For all a, b ∈ G, a · b = b · a [commutativity]. Example 0.2.31 (Ring). A ring is an algebra hA, +, ·, −, 0i such that the operations +, ·, − and 0 are binary, binary, unary and nullary, respectively and the following conditions are satisfied: (A1) hA, +, −, 0i is an abelian group; (A2) hA, ·i is a semigroup; (A3) For all a, b, c ∈ A, a · (b + c) = (a · b) + (a · c) and (a + b) · c = (a · c) + (b · c) [distributivity]. Example 0.2.32 (Field). A field is an algebra hC, +, ·, 1, 0i such that the operations + and · are binary, the operations 1 and 0 are nullary, and the following conditions are satisfied:


25

(C1) hC, +, −, 0i is an abelian group; (C2) hC − {0}, ·, −1 , 1i is an abelian group; (C3) For all a, b, c ∈ C, a · (b + c) = a · b + a · c [distributivity]. Example 0.2.33 (Semilattice). A semigroup hG, ·i is called of semilattice if it satisfies the following identities: (G4) For all a, b ∈ G, a · b = b · a [commutativity]; (S2) For all a ∈ G, a · a = a [idempotency]. Example 0.2.34 (Lattice). A lattice is an algebra hL, ∧, ∨i such that ∧ and ∨ are binary operations and the following identities are satisfied: (L1) For all a ∈ L, a ∧ a = a and a ∨ a = a [idempotency] (L2) For all a, b, c ∈ L, (a ∧ b) ∧ c = a ∧ (b ∧ c) and (a ∨ b) ∨ c = a ∨ (b ∨ c) [associativity] (L3) For all a, b ∈ L, a ∧ b = b ∧ a and a ∨ b = b ∨ a [commutativity] (L4) For all a, b ∈ L, (a ∧ b) ∨ b = b and (a ∨ b) ∧ b = b [absorption]. Example 0.2.35 (Distributive lattice). A lattice hL, ∧, ∨i is distributive if the following identities are satisfied: (L4) For all a, b, c ∈ L, (a ∨ b) ∧ c = (a ∧ c) ∨ (b ∧ c) and (a ∧ b) ∨ c = (a ∨ c) ∧ (b ∨ c) [distributivity]. Example 0.2.36 (Bounded lattice). A lattice hL, ∧, ∨i is said to be bounded from below if there is an element 0 ∈ L, such that 0 ≤ a for all a ∈ L and L is said to be bounded from above if there is an element 1 ∈ L, such that a ≤ 1 for all a ∈ L. A bounded lattice is one that is bounded both from above and below. Example 0.2.37 (Boolean algebra). A Boolean algebra is an algebra hB, ∧, ∨,0 , 0, 1i such that ∧ and ∨ are binary operations, 0 is an unary operation, 0 and 1 are nullary operations and the following identities are satisfied: (B1) hB, ∧, ∨i is a distributive lattice; (B2) For all a ∈ B, a ∧ 0 = 0 and a ∨ 1 = 1; (B3) For all a ∈ B, a ∧ a0 = 0 and a ∨ a0 = 1 [complement].


26

Example 0.2.38 (Heyting algebra). A Heyting algebra is an algebra hH, ∧, ∨, →, 0, 1i, such that ∧, ∨ e → are binary operations, 0 and 1 are nullary operations and the following identities are satisfied: (H1) hH, ∧, ∨i is a distributive lattice; (H2) For all a ∈ H, a ∧ 0 = 0 and a ∨ 1 = 1; (H3) For all a ∈ H, a → a = 1; (H4) For all a, b ∈ H, (a → b) ∧ b = b and a ∧ (a → b) = a ∧ b; (H5) For all a, b, c ∈ H, a → (b∧c) = (a → b)∧(a → c) and (a∨b) → c = (a → c)∧(b → c). Example 0.2.39 (Heyting algebra). A Heyting algebra is a bounded lattice (whose maximum and minimum elements are denoted 1 and 0, respectively) with a binary operation → such that a ∧ c ≤ b if and only if c ≤ (a → b). Proposition 0.2.40. The two definitions of Heyting algebras above (Examples 0.2.38 and 0.2.39) are equivalents. The two following results are well known in the literature. A proof can be found in (RASIOWA; SIKORSKI, 1963). Proposition 0.2.41. Let hH, ∧, ∨, 0, 1i be a lattice satisfying (H1) and (H2) of the Definition 0.2.38, if there exists → such that (H3) - (H5) of the Definition 0.2.38 are true, then → is unique. Proposition 0.2.42. If A = hA, ∧, ∨, →, 0, 1i is a Heyting algebra, then A is a Boolean algebra if and only if a ∨ (a → b) = 1, for any a, b ∈ A. Definition 0.2.43 (Order lattice). An order lattice is a poset hL, ≤i such that for every a, b ∈ L both sup{a, b} and inf {a, b} exist. Remark 0.2.44. The two lattice definitions (see Example 0.2.34 and Definition 0.2.43) are equivalent. To show it, if L is a lattice by Example 0.2.34, then we define ≤ over L by a ≤ b iff 3 a = a ∧ b (or by a ≤ b iff b = a ∨ b) and when L is an order lattice, then we define the operations ∧ and ∨ by a ∧ b = inf {a, b} and a ∨ b = sup{a, b}. For more details about this proof, see (BURRIS; SANKAPPANAVAR, 1981, p. 8). Definition 0.2.45. An implicative lattice is an algebra A = hA, ∧, ∨, →, 1i where W hA, ∧, ∨, 1i is a lattice with the greatest element 1, such that {c ∈ A : a∧c ≤ b} exists for 3

Abbreviation for “if and only if”.


27

every a, b ∈ A,4 and → is the induced implication given by a → b = {c ∈ A : a ∧ c ≤ b} def for every a, b ∈ A (note that 1 = a → a is the top element of A, for any a ∈ A). If, additionally, a ∨ (a → b) = 1 for every a, b then A is said to be a classical implicative lattice.5 W

The following well known results connect implicative lattice with Heyting algebra and classical implicative lattice with Boolean algebra: Proposition 0.2.46. Let A be an implicative lattice. Then: (1) If A has a bottom element 0, then it is a Heyting algebra. (2) If A is a classical implicative lattice and it has a bottom element 0, then it is a Boolean algebra.

0.3 Notions of category theory In this section, we will introduce the basic notions of category theory that will be used in this Thesis, mainly in the Section 2.5 and in the Chapter 4. The subject presented here can be found in any introductory book about category theory. In particular, we suggest the book “Category Theory” (AWODEY, 2010). Definition 0.3.1 (Category). A category C is given by a collection Ob(C) of elements A, B, C, D, . . ., called objects of C and a collection M or(C) of elements f, g, h, . . ., called arrows or morphisms of C that satisfy the following conditions: (i) For each morphism f of C is assigned a pair of objects dom(f ), cod(f ) called domain of f and codomain of f , respectively. If A = dom(f ) and B = cod(f ), we say that f is a morphism from A to B and we denote it by f : A → B; (ii) For each pair of morphisms f : A → B, g : B → C of C is assigned a morphism g ◦ f : A → C called composition of f and g; (iii) For each object A of C is assigned a morphism 1A : A → A, called identity morphism of A; (iv) If f : A → B, g : B → C and h : C → D are morphisms, then h ◦ (g ◦ f ) = (h ◦ g) ◦ f (associativity); (v) For each object B of C, let f : A → B and g : B → A be morphisms, we have 1B ◦ f = f and g ◦ 1B = g (identity); 4

5

W Here, X denotes the supremum of the set X ⊆ A w.r.t. partial order associated with the lattice ≤, provided that it exists. The name was taken from H. Curry, see (CURRY, 1977).


28

(vi) For any objects A, B of C, the collection M orC (A, B) of the morphisms from A to B is a set. Definition 0.3.2 (Subcategory). The category D is a subcategory of the category C if: (i) Ob(D) ⊆ Ob(C); (ii) If A, B ∈ Ob(D), then M orD (A, B) ⊆ M orC (A, B). Definition 0.3.3 (Full subcategory). The category D is a full subcategory of the category C if: (i) D is a subcategory of C; (ii) If A, B ∈ Ob(D), then M orD (A, B) = M orC (A, B). Definition 0.3.4 (Diagram). A diagram in a category C is a (possibly empty) set of objects of C with a (possibly empty) set of morphisms between these objects. The objects of a diagram C are called vertices of C and the morphisms are called edges of C. Definition 0.3.5 (Branch). Let C be a category. A branch in C is a finite sequence f1 , . . . , fn of edges of C, such that dom(fi+1 ) = cod(fi ) for i ∈ {1, 2, . . . , n − 1}. Definition 0.3.6 (Commutative diagram). Let C be a category. A diagram is commutative (or the diagram commutes) if, for any branches f1 , . . . , fn and g1 , . . . , gm in C, with n ≥ 2 or m ≥ 2, we have fn ◦ fn−1 ◦ . . . ◦ f1 = gm ◦ gm−1 ◦ . . . ◦ g1 . Definition 0.3.7 (Monomorphism). The morphism f : A → B of a category C is a monomorphism if for every pair of morphisms g : C → A, h : C → A of C, if f ◦ g = f ◦ h, then g = h. Definition 0.3.8 (Epimorphism). The morphism f : A → B of a category C is an epimorphism if for every pair of morphisms g : B → C, h : B → C of C, if g ◦ f = h ◦ f , then g = h. Definition 0.3.9 (Isomorphism). The morphism f : A → B of a category C is an isomorphism if there is a morphisms g : B → A in C, such that g ◦ f = 1A and f ◦ g = 1B . Definition 0.3.10 (Initial object). Let C be a category. Um object 0 is initial if for each object B ∈ Ob(C) there is a unique morphism f : 0 → B in C. Definition 0.3.11 (Terminal object). Let C be a category. Um object 1 is terminal if for each object C ∈ Ob(C) there is a unique morphism g : C → 1 in C.


29

Definition 0.3.12 (Product). Let C be a category and A, B ∈ Ob(C). A product of the objects A and B in C is hA × B, {pA , pB }i where A × B is an object of C and pA : A × B → A, pB : A × B → B are morphisms of C such that, for every pair of morphisms f : C → A, g : C → B of C, there is a unique morphism from C to A × B such that the diagram below commutes: C

f

{

g

A×B pA

pB

#

A

B

Definition 0.3.13 (Product of a family). Let C be a category and {Ai }i∈I a family of Q Q objects of C. A product of the family {Ai }i∈I in C is h i∈I Ai , {pi }i∈I i where i∈I Ai is Q an object of C and pi : i∈I Ai → Ai are morphisms of C such that if fi : C → Ai is a Q family of morphisms of C, then there is a unique morphism from C to i∈I Ai such that the diagram below commutes (for any i ∈ I): C fi

Q

i∈I

Ai

# pi

/

Ai

Definition 0.3.14 (Product in the category). A category C has (finite) products if there is, in C, the (finite) product of any set of objects of C. Definition 0.3.15 (Functor). Let C and D be two categories. A functor F from C to D, denoted F : C → D, is a map that assigns: (i) For each object A ∈ Ob(C), an object F (A) ∈ Ob(D); (ii) For each morphism f ∈ M orC (A, B), a morphism F (f ) ∈ M orD (F (A), F (B)) that satisfies: (a) F (f ◦ g) = F (f ) ◦ F (g), if dom(f ) = cod(g) for every pair of morphisms f, g ∈ C; (b) F (1A ) = 1F (A) , for every A ∈ Ob(C).

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1 Algebraic Hyperstructures: Origins

This chapter is exclusively focused on the historical chronology, no technical developments (relationship between several definitions of the same concept, for instance) will be done here. Doing this would constitute the subject of a research project of a different nature than the present thesis. An operation is basically a function1 that manipulate elements of a set and returns a value that is in another set. And if we think about a more general definition? Something that returns a set of values instead of a single value? This concept has already been thought and named multioperation (or hyperoperation). “Operation” is a fundamental concept in algebraic theory. It is used to define algebras, that are structures such as groups, lattices, rings2 , composed by a set and at least one operation. In a similar way, we can have structures composed by at least one multioperation and the class of this kind of structures is what we call of algebraic hyperstructure. The elements in the class of hyperstructures are multialgebras, that is algebras equipped with at least one multioperation. In this chapter we will present some examples of multialgebras, such as multigroups, multilattices and so on, from a historical point of view and we will relate these hyperstructures with non-deterministic matrices. The concept of non-deterministic matrices will be presented with more rigor in the last section, but in essence, non-deterministic matrices are multialgebras and they have been used as semantics for logical systems that do not have usual deterministic semantics. As we will see, hyperstructures have been studied by several authors, but with different nomenclatures and this hinders the development of a detailed and precise chronology. The intrinsic relationship between multialgebras and non-deterministic matrices is not evident in the literature. In this sense, we believe that our work will give an important contribution. 1 2

The mathematical concept in the previous chapter. For a formal definition of these concepts, see the previous chapter.

Chapter 1. Algebraic Hyperstructures: Origins

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1.1 Hypergroups The study of hyperstructures began with the presentation of the paper entitled “Sur une généralisation de la notion de groupe”, in 1934 by the French mathematician Frédéric Marty in the 8º Congress of Scandinavian Mathematicians (MARTY, 1934). In this paper, Marty presents the notion of hypergroups (or multigroups) from the analysis of their properties. But, due to his premature death, Marty only published two papers related to his concept of hypergroups. 1934 was the year that Frederic Marty defined the hypergroup (MARTY, 1934). This happened in connection with his thesis on meromorphic functions, which was written under the direction of Paul Montel. Unfortunately F. Marty died young, during the Second World War, when his airplane was shot down over the Baltic Sea, while he was going on a mission to Finland. In the duration of his short life (1911-1940), F. Marty studied properties and applications of the hypergroups in two more communications (MARTY, 1935; MARTY, 1936). (MASSOUROS; MASSOUROS, 2006, p. 19)

In 1937, H. S. Wall (WALL, 1937) and M. Krasner (KRASNER, 1937) also gave their respective definitions of hypergroup. As R. Bayon and N. Lygeros highlight in (BAYON; LYGEROS, 2008, p. 821), the origin of the hyperstructures is still not completely known but the Marty’s hypergroup definition took to emergence of several works related to multialgebras. We will quote some of them in the course of this chapter. In (MARTY, 1934), Frédéric Marty introduced the following definition of hypergroup: Let be a set of elements, non-empty, with four combination laws: AB, BA, A A B| and |B , each of which may have several determinations; the first two are associative. If C is a determination of AB we write AB ⊃ C, (AB contains C). We will say that the family is a hypergroup if the two divisions are A ⊃ C BC ⊃ A and related to multiplication by the following relations: B| A 3 |B ⊃ C CB ⊃ A (MARTY, 1934, p. 46, our translation)

In the definition of F. Marty, represents the division on the left.

A B|

represents the division on the right and

A |B

The definition of Marty is equivalent to the following definition: Definition 1.1.1. (KRASNER, 1937) A set H organized by a composition law a.b of each pair a, b ∈ H is called hypergroup with regard to this composition law if 3

A A Soit un ensemble d’éléments, non vide, possédant quatre lois de combinaison AB, BA, B| et |B ; chacune d’elles pouvant avoir plusieurs déterminations; les deux premières étant associatives. Si C est une détermination de AB nous écrirons AB ⊃ C, (AB contient C). Nous dirons que la famille constitue un hypergroupe si les deux divisions sont liées à la multiplication par les relations suivantes: A A B| ⊃ C BC ⊃ A et |B ⊃ C CB ⊃ A (MARTY, 1934, p. 46)


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(i) a.b is a non-empty subset of H; (ii) (a.b).c = a(b.c) (associative law); (iii) For each pair a, c ∈ H there is x ∈ H such that c ∈ a.x and there is x0 ∈ H such that c ∈ x0 .a. Also Christos G. Massouros, in his paper (MASSOUROS, 1989, p. 7) shows how to derive the property of regenerativity4 from original definition of F. Marty and later on, in (MASSOUROS; MASSOUROS, 2006; MASSOUROS; MASSOUROS, 2014) the authors present the following definition of hypergroup equivalent to the definition of F. Marty. Remark 1.1.2. Many mathematicians in several countries contributed to the studies of the hypergroups, a bibliographical list can be found online on the site of A.H.A. (Algebraic Hyperstructures and Applications)5 , a scientific group of Democritus University of Thrace in Greece. One of the first books dedicated to hypergroups was written by P. Corsini in 1993 and it is named “Prolegomena of hypergroup theory” (CORSINI, 1993). In the following, we will present another important definition in the history of the hyperstructures: the hyperlattice definition.

1.2 Hyperlattices The concept of hyperlattice was introduced by the Romanian algebraist Mihail Benado in 1953 in the paper “Asupra unei generalizˇari a noţiunii de structurˇa” (BENADO, 1953). In this work, Benado presents two equivalent definitions of hyperlattice and also some examples. Despite the introduction of the concept already appears in the work of Benado of 1953, several authors uses the paper “Les ensembles partiellement ordonnés et le théorème de raffinement de Schreier. II. Théorie des multistructures” (BENADO, 1955), published in 1955 as initial reference for the hyperlattice theory. In this work, Benado called théorie des multistructures to what we call of hyperlattice theory. In this section our definition will be based on (BENADO, 1955). Definition 1.2.1. (BENADO, 1955) A multistructure 6 (now known as multilattice or hyperlattice) is any partially ordered set (poset)7 P that satisfies the following principles: 4 5 6

7

Let hH, ·i be a hypergroup and a ∈ H, the regenerativity is the property: a.H = H.a = H. . Remember that in this work, we use the name multistructure/hyperstructure to denote the class of all multialgebras such as hypergroups, hyperlattices, and so on. See Definition 0.1.11.


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M 1) Let a, b ∈ P , if there is Ω ∈ P such that Ω ≥ a and Ω ≥ b, then there is also an M ∈ P such that M ≤ Ω, M ≥ a and M ≥ b and the conditions x ≤ M , x ≥ a and x ≥ b imply x = M . M 2) Let a, b ∈ P , if there is ω ∈ P such that ω ≤ a and ω ≤ b, then there is also a d ∈ P such that d ≥ ω, d ≤ a and d ≤ b and the conditions y ≥ d, y ≤ a and y ≤ b imply y = d. Benado uses the notation (a ∨ b)Ω to denote all M of (M 1), that is (a ∨ b)Ω = {M : M is minimal in {x : x ≥ a, x ≥ b} and M ≤ Ω} and he uses the notation (a ∧ b)ω to denote all d of (M 2), that is (a ∧ b)ω = {d : d is maximal in {y : y ≤ a, y ≤ b} and d ≥ ω}. [. . .] I see that the meaning and the validity of several fundamental principles of the structures theory do not depend on the fact that in the structure S, a ∨ b and a ∧ b are respectively the upper and lower bounds [12]8 of elements a, b ∈ S, but it depends only on the fact that a ∨ b and a ∧ b are respectively a minimal element [12] between all x ∈ S, such that Ω ≥ x ≥ a, b and a maximal element [12] between all y ∈ S, such that ω ≤ y ≤ a, b (here Ω, ω are arbitrary, but fixed) (BENADO, 1955, p. 309, our translation)9 .

In this citation, the author makes clear the difference between lattices and hyperlattices. The main difference between Benado’s definition and the usual definition of supremum (infimum, respectively) is that the minimal upper bounds (maximal lower bounds, resp.) are considered instead of the minimum (the maximum, resp.)10 Still in the paper (BENADO, 1955), Benado presents also his definition of submultistructure (or subhyperlattice as we call here). Definition 1.2.2. (BENADO, 1955) Let M be any multistructure, a non-empty subset R of M is a submultistructure of M if R satisfies the following conditions: 10 ) If a, b ∈ R and there is at least an Ω ∈ R such that Ω ≥ a and Ω ≥ b, then R ∩ (a ∨ b)Ω 6= ∅. 20 ) If a, b ∈ R and there is at least an ω ∈ R such that ω ≤ a and ω ≤ b, then R ∩ (a ∧ b)ω 6= ∅. 8

9

10

By [12], Benado refers to “N. Bourbaki, Théorie des ensembles (fasc. de résultats), Paris, Hermann, 1939”. [. . .] je me suis rendu compte de ce que le sens et la validité de plusieurs principes fondamentaux de la théorie des structures, ne dépendent nullement du fait que, dans la structure S, a ∨ b et a ∧ b sont respectivement les bornes supérieure et inférieure [12] des éléments a, b ∈ S; mais ils dépendent uniquement du fait que a ∨ b et a ∧ b y sont respectivement un élément minimal [12] parmi tous les x ∈ S tels que Ω ≥ x ≥ a, b et un élément maximal [12] parmi tous les y ∈ S tels que ω ≤ y ≤ a, b (ici Ω, ω sont arbitraires mais fixes)(BENADO, 1955, p. 309). For these concepts, see the Chapter 0.


34

Classically, a subset R0 is a sublattice of a lattice R if R0 is closed under the operations ∨ and ∧, that is, for every a, b ∈ R0 , (a ∨ b) ∈ R0 and (a ∧ b) ∈ R0 . Similarly, the definition of Benado provides that to R be a submultistructure of M, for every a, b ∈ R, R must have at least one element that satisfies the condition (M 1) and at least one element that satisfies the condition (M 2) (of multistructure definition). And yet, Benado (BENADO, 1955) says that a submultistructure R of M is closed if, for every a, b ∈ R of (10 ) we have that (a ∨ b)Ω ⊆ R and for each a, b ∈ R of (20 ) we have that (a ∧ b)ω ⊆ R. Definition 1.2.3. (BENADO, 1955, p. 321) A multistructure is a non-empty set R with two operations ∧ and ∨11 satisfying the following axioms: M I) Let a, b ∈ R, if a ∨ b 6= ∅ (a ∧ b 6= ∅) and b ∨ a 6= ∅ (b ∧ a 6= ∅). Then: I 0 ) a ∨ b = b ∨ a; I 00 ) a ∧ b = b ∧ a. M II) Let a, b, c ∈ R, if a ∨ b = 6 ∅ and (a ∨ b) ∨ c 6= ∅ (a ∧ b = 6 ∅ and (a ∧ b) ∧ c 6= ∅) and if b ∨ c 6= ∅ and a ∨ (b ∨ c) 6= ∅ (b ∧ c 6= ∅ and a ∧ (b ∧ c) 6= ∅), then II 0 ) for each M ∈ (a ∨ b) ∨ c there is an M 0 ∈ a ∨ (b ∨ c) such that M ∨ M 0 6= ∅ and M ∨ M0 = M; II 00 ) for each d ∈ (a ∧ b) ∧ c there is a d0 ∈ a ∧ (b ∧ c) such that d ∧ d0 = 6 ∅ and d ∧ d0 = d. M III) Let a, b ∈ R, if a ∨ b 6= ∅ (a ∧ b 6= ∅) and if a ∧ (a ∨ b) 6= ∅ (a ∨ (a ∧ b) 6= ∅) then III 0 ) a ∧ (a ∨ b) = a; III 00 ) a ∨ (a ∧ b) = a. M IV ) For each a ∈ R we have a ∨ a 6= ∅ and a ∧ a 6= ∅. M V ) Let a, b, c ∈ R such that a = b and if c ∨ a = 6 ∅ (c ∧ a 6= ∅) and if c ∨ b 6= ∅ (c ∧ b 6= ∅) then V 0 ) c ∨ a = c ∨ b; V 00 ) c ∧ a = c ∧ b. M V I) Let a, b ∈ R such that a∨b = 6 ∅ (a∧b 6= ∅) and let M, M 0 ∈ a∨b (d, d0 ∈ a∧b) such that M ∨ M 0 6= ∅ (d ∧ d0 6= ∅) then if M = 6 M 0 (d = 6 d0 ) we have M 00 6= M, M 0 (d00 6= d, d0 ) for each M 00 ∈ M ∨ M 0 (d00 ∈ d ∧ d0 ). 11

According to Benado (BENADO, 1955) these operations are not necessarily universal and unambiguous, that is, there are a, b ∈ R, such that if a ∨ b 6= ∅ (a ∧ b 6= ∅), then the set a ∨ b (a ∧ b) may has at least two distinct elements.


35

The axiom (M I) refers to commutativity, the axiom (M II) is a kind of partial associativity. Marty called the axiom (M III) of reduction (usually we call absorption). The axiom (M IV ) only says that a ∨ a and a ∧ a are not empty (for every a ∈ R), while the axiom (M V ) preserves equality between c ∨ a = c ∨ b and c ∧ a = c ∧ b in the case of a = b. Benado (BENADO, 1955) also showed that (M I)-(M V I) of Definition 1.2.3 and (M 1)-(M 2) of Definition 1.2.1 are equivalent. Hyperlattices have also been studied by other authors as D.J. Hansen (HANSEN, 1981), that presents an alternative to the axiomatization given by Benado for characterization of a hyperlattice. The motivation of Hansen is to avoid partial associativity in Benado definition. The new axiomatic of Hansen only validates the axioms (M I) and (M III) above and adds three new axioms. Also in (MARTíNEZ et al., 2001), we found an alternative definition of hyperlattice which aims to eliminate some disadvantages generated by generalized associativity in hyperlattice definitions of Benado and Hansen. Among the disadvantages cited by the authors, we have a non natural generalization of the associative property and the fact that such properties do not allow algebraic definition of submultilattice. So, (MARTíNEZ et al., 2001) introduce a new algebraic structure of hyperlattice with a weaker associative property. After Benado and before Hansen, the Brazilian mathematician Antonio Antunes Mario Sette, motivated by algebraization of Cω 12 , introduced the concept of hyperlattice Cω (SETTE, 1971) in his Master’s thesis (1971) supervised by Professor Newton da Costa. To introduce the concept of hyperlattice Cω , Sette used the definition of hyperlattice presented by José Morgado in the book “Introdução à Teoria dos Reticulados” (MORGADO, 1962). Morgado calls his hyperlattices of “reticuloides” and he uses the concepts of “supremoide” and “infimoide” in his definition. Definition 1.2.4. (MORGADO, 1962; SETTE, 1971) A hyperlattice (reticuloide) is a system hR, 6i consisting of a set R 6= ∅ and a quasi-order ≤ (relation only reflexive and transitive) such that, for all a, b ∈ R, a 4 b 6= ∅ = 6 a5b where a4b (infimoide) is the set of all infimum of the pair (a, b) ∈ R2 and a5b (supremoide) is the set of all supremum of the pair (a, b) ∈ R2 . Remark 1.2.5. Note that in a partially ordered system, by antisymmetric property, we can show the uniqueness of the supremum (infimum) (if it exists), but this result is not 12

Cω is one of the paraconsistent logics wich is part of the systems Cn (1 6 n 6 ω) of Newton da Costa (COSTA, 1963; COSTA, 1974).


36

obtained in the quasi-ordered systems13 . Since to have more than one supremum (infimum) it is possible, it is legitimate to define the supremoide (infimoide) as the set of all suprema (infima). Morgado also presents another reticuloide definition: Definition 1.2.6. (MORGADO, 1962) A hyperlattice (reticuloide) is a system hR, 4, 5i consisting of a set R = 6 ∅ and two operations 4, 5 : R × R −→ (P(R) − {∅}) such that, for all a, b, c ∈ R, the following conditions are satisfied: h1) a 5 b = b 5 a; h2) If x ∈ a 5 b and y ∈ b 5 c, then x 5 c = a 5 y; h3) If x ∈ a 5 b, then a ∈ a 4 x. h4) a 4 b = b 4 a; h5) If x ∈ a 4 b and y ∈ b 4 c, then x 4 c = a 4 y; h6) If x ∈ a 4 b, then a ∈ a 5 x; The definitions of Morgado are more intuitive (more similar that usual lattice definition) than those by Benado. In the Definition 1.2.4 what changes in relation to the usual definition, is the loss of the uniqueness of the supremum and of the infimum. And in Definition 1.2.6, similar to the usual case, the items (h1) and (h4) correspond to generalization of the commutativity, the items (h2) and (h5) are a kind of associativity and the items (h3) and (h6) are a kind of generalization of the absorption property. In Morgado’s book and in the Sette’s dissertation we found only some considerations about the hyperlattices, a definition of an implicative hyperlattice and a definition of the hyperlattice Cω . Sette, in the concluding remarks, notes that algebraization of inconsistent formal systems can lead us to the consideration of hypersystems, this could be a generalization of reticuloides (hyperlattices), but he did not give any other information about the development of such hypersystems. The generalization of this idea of hypergroup and hyperlattice led to the emergence of hyperalgebras/multialgebras. These more general structures will be the topic of the next section.

1.3 Multialgebras 13

That is the systems composed by a set and a quasi-order. For more information about these concepts, see Chapter 0.


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In general, multialgebras (also known as hyperalgebras) are algebras such that the operations can return, for a given entry, a set of values instead of a single value. The origin of multialgebras is a little obscure, because of a very large number of papers came from Marty’s paper in 1934 (MARTY, 1934). Some author, for example (WALICKI, 2005), consider as seminal work the two papers: “Algebras with Operators. Part I” (JONSSON; TARSKI, 1951) and “Algebras with Operators” (JONSSON; TARSKI, 1952), both published by Bjarni Jonsson and Alfred Tarski. In these papers, Jonsson and Tarski introduce the concept of complex algebra and they prove the representation of Boolean algebras with operators by means of these algebras. However, the term “complex algebra” has several meanings in the literature and is usually found in authors such as (C¯IRULIS, 2004) and (BOŠNJAK; MADARÁSZ, 2003), with the name of full complex algebra. Definition 1.3.1. (JONSSON; TARSKI, 1951, p. 933) A complex algebra of a relational structure14 U = hU, {Ri }i∈I i is defined by: U + = hP(U ), {Ri+ }i∈I i such that, (i) U + is a Boolean algebra with operators. A Boolean algebra with operators (BAO) is an algebra hA, {fi }i∈I i such that A is a Boolean algebra and fi are operators, that is, operations over the Boolean algebra that are additive (distributive on the usual Boolean addition) on each of its arguments; (ii) Let R ⊆ U n+1 , then Ri+ : P(U )n −→ P(U ) is an operation such that R+ (X0 , X1 , . . . , Xn−1 ) = {y ∈ U : (x0 , x1 , . . . , xn−1 , y) ∈ R, for x0 ∈ X0 , x1 ∈ X1 , . . . , xn−1 ∈ Xn−1 }. Following Marty and Benados’ line, that is, by defining hyperalgebras from multioperations, we can say that the origin of multialgebras can be found in the paper “Bemerkungen zum Homomorphiebegriff” (Comments to the concept of homomorphism) of Günter Pickert, published in 1950 (PICKERT, 1950). The goal of the author in this paper was to define homomorphism from structures, but what he calls “structure” is what we call multialgebra. In 1958, and probably independently, the concept of multialgebra was introduced by P. Brunovský in “O zovšeobecnených algebraických systémoch” (A generalization of algebraic systems) (BRUNOVSKý, 1958). Brunovský’s definitions of multioperation and multialgebra are the following: 14

In (JONSSON; TARSKI, 1951, p. 933), the authors use the term algebra in the wider sense.


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Definition 1.3.2. (BRUNOVSKý, 1958) An n-ary generalized operation fα in the set A is a function, such that for all sequences of elements n = n(α) in A, assigns any subset of the set A. Definition 1.3.3. (BRUNOVSKý, 1958) Let be A a set and F any set of generalized operations in A. The set A with the set F will be designated multialgebra of A. Brunovský cites the Benado multilattices as an example of multialgebras. Note that the definition of Brunovský is quite similar to the definition of more recent authors like Hansoul, Schweigert and Ameri and Rosenberg. Definition 1.3.4. (HANSOUL, 1981; SCHWEIGERT, 1985; AMERI; ROSENBERG, 2009) Let A be a non-empty set, a multioperation (or hyperoperation) (n-ary) σ on A is a function σ : An → (P(A) − {∅}), such that n is a positive integer. Definition 1.3.5. (HANSOUL, 1981; SCHWEIGERT, 1985; AMERI; ROSENBERG, 2009) A multialgebra (or hyperalgebra) is the pair hA, {σi }i∈I i, such that A is a non-empty set and {σi }i∈I is a family of multioperations on A. The main difference between the Definitions 1.3.2, 1.3.3 and 1.3.4, 1.3.5 is that the value of an operation to be empty. Multialgebras can be defined as relational structures with a composition of relations of arbitrary arity. Properties of multialgebras seen as relational systems can be found in (PICKETT, 1964). We present, now, some important concepts in the multialgebras theory.

1.3.1 Homomorphisms of multialgebras and other concepts There are in the literature several generalizations of the notion of homomorphism for multialgebras. We detected that Marty, in his paper “Rôle de la notion d’hypergroupe dans l’étude des groupes non abéliens” (MARTY, 1935) already presented a concept of homomorphism for hypergroups: “[. . .] a representation of a hypergroup over (or in) other is a homomorphism if the image of a determination of the product is the determination of the product of the images.”15 (MARTY, 1935, p. 636, our translation). The definition of Marty means that given hG1 , ·1 i and hG2 , ·2 i two hypergroups16 , the function h from G1 to G2 is a hypergroups homomorphism if, for all x, x ∈ a.1 b ⇒ h(x) ∈ (h(a).2 h(b)) 15

16

The original in French: “[. . .] une représentation d’une hypergroupe sur (ou dans) un autre est une homomorphie si l’image d’une détermination du produit est détermination du produit des images”. According to the definition of Marty (MARTY, 1934).


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. Marty says, in the same paper, that a isomorphism between hypergroups is a homomorphism such that the correspondence (in the definition above) is biunivocal. The author also remarks the necessity to distinguish degrees of homomorphism and he presents his definition of a quasi isomorphism, which is basically a kind of surjective homomorphism between hypergroups. For the generalized notion of the multialgebras, again we can say that Brunovský, in (BRUNOVSKý, 1958), was the first to introduce a definition of homomorphism for multialgebras17 . Definition 1.3.6. (BRUNOVSKý, 1958) Let A = hA, F i and B = hB, Gi be two multialgebras of same type18 and let h be a function from A to B. Then, h is said to be a homomorphism if, for all n-ary generalized operations fα ∈ F and for any sequence of elements x1 , . . . , xn in A, it is true: h[fα (x1 , . . . , xn )] = fα [h(x1 ), . . . , h(xn )]. In the literature, however, there are several definitions of homomorphism between multialgebras. In 1979, Francis Maurice Nolan, in his Ph.D. Thesis (NOLAN, 1979), introduced five definitions of multialgebras homomorphisms and constructed a category to each one. The definitions of Nolan are the following: Definition 1.3.7. Let A = hA, F i and B = hB, F 0 i be two multialgebras of same type and let h be a function from A to B, • h is a full homomorphism between the multialgebras A and B if, for every f 0 ∈ F 0 , for every a ∈ A and for any sequence b1 , . . . , bn ∈ B, h(a) ∈ f 0 (b1 , . . . , bn ) iff there are a1 , . . . , an ∈ A such that a ∈ f (a1 , . . . , an ) and h(ai ) = bi for every i such that 1 ≤ i ≤ n. • h is a weak homomorphism between the multialgebras A and B if, for every f ∈ F and for any sequence a1 , . . . , an ∈ A, h(f (a1 , . . . , an )) ⊆ f 0 (h(a1 ), . . . , h(an )). • h is a strong homomorphism between the multialgebras A and B if, for every f ∈ F and for any sequence a1 , . . . , an ∈ A, h(f (a1 , . . . , an )) = f 0 (h(a1 ), . . . , h(an )). 17 18

Multialgebras in the sense of Definitions 1.3.2 and 1.3.3. That is, let A = hA, F i and B = hB, Gi be two multialgebras such that F and G are their respective sets of n-ary generalized operations. We say that the multialgebras A and B are of same type if it is possible to establish a biunivocal correspondence between the n-ary generalized operations of A and the n-ary generalized operations of B, such that for each operation fα ∈ F , the operation fβ ∈ G corresponding to it, will be n-ary with the same n.


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• h is a bimorphism between the multialgebras A and B if, for every f 0 ∈ F 0 , for every a ∈ A and for any sequence h(a1 ), . . . , h(an ) ∈ h[A], h(a) ∈ f 0 (h(a1 ), . . . , h(an )) iff a ∈ f (a1 , . . . , an ). • h is an absolute homomorphism between the multialgebras A and B if, for every f 0 ∈ F 0 , for every b ∈ B and for any sequence b1 , . . . , bn ∈ B, b ∈ f 0 (b1 , . . . , bn ) iff there are a, a1 , . . . , an ∈ A such that a ∈ f (a1 , . . . , an ), h(a) = b and h(ai ) = bi for every i such that 1 ≤ i ≤ n. Similarly to concept of homomorphism, other concepts such as congruence, submultialgebra, direct product and so on, can also be defined in the context of the multialgebras. These concepts will be formally introduced in the next chapter and, therefore, we will talk briefly about the origin of some of them. The concept of congruence was defined in the framework of multialgebras by Schweigert, in the paper called “Congruence relations of multialgebras”, published in 1985 (SCHWEIGERT, 1985). The goal of Schweigert was to find a suitable concept of variety of multialgebras. In this paper, Schweigert says that the Birkhoff theorem19 is valid for multialgebras. However, the author simply notes that the demonstration is similar to the statement of results for the usual algebras. The problem with this is that, as we saw before, in multialgebras we have many possibilities to define concepts such as homomorphism, submultialgebras, and others. Being so, this ambiguity is not a minor issue. The concept of congruence of multialgebras is studied in detail in the paper “Congruences of multialgebras” of Reza Ameri and Ivo G. Rosenberg (AMERI; ROSENBERG, 2009). Other concepts such as identities and direct limit of multialgebras can be found on the Ph.D. Thesis by Cosmin Pelea (PELEA, 2003). In the next section, we will briefly discuss an important result for the multialgebras theory.

1.3.2 Representation theorem The result known as representation theorem for multialgebras was studied firstly in (GRÄTZER, 1962) and (HöFT; HOWARD, 1981). The representation theorem is one of the most important results on multialgebras theory. Intuitively speaking, it proves that the study of multialgebras is a natural extension of the theory of universal algebra. This theorem was introduced by G. Grätzer, in the paper entitled “A representation theorem for multi-algebras” (GRÄTZER, 1962). But this theorem does not apply to multialgebras with 19

All multialgebra can be represented as a product of directly irreducible multialgebras


41

multioperations such that the images can be an empty set, because these multialgebras are closer to relational systems than to universal algebra. The representation theorem of Grätzer uses the concept of concrete multialgebra. Definition 1.3.8. (GRÄTZER, 1962, p. 453) Let A be an algebra with domain A and a collection of operations F , and let θ be an equivalence relation on A. A concrete multialgebra is a multialgebra A/θ which consists of the following elements: (i) a set A/θ of the equivalence classes, such that if a ∈ A, then a/θ is the equivalence class represented by a; and (ii) a set of multioperations of kind f (a1 /θ, . . . , an /θ), such that if f ∈ F , then each multioperation is defined by: f (a1 /θ, . . . , an /θ) = {f (b1 , . . . , bn )/θ : b1 ∈ a1 /θ, . . . , bn ∈ an /θ}. Theorem 1.3.9. (GRÄTZER, 1962, p. 453) Every multialgebra is concrete. This representation theorem states that every multialgebra is concrete, what means that, for example, if A is a multialgebra with the only binary multioperation ·, then there is an algebra B with a binary operation · and with a equivalence relation θ such that B/θ (defined as above) is isomorphic to A. So, every multialgebra has an algebra that represents it. Grätzer, also in his paper of 1962, lists some problems that arise naturally from the representation theorem. For example, is the theorem equivalent to the axiom of choice? (Problem 1). This problem was solved by H. Höft and P. E. Howard in the paper “Representing multi-algebras by algebras, the axiom of choice, and the axiom of dependent choice” (HöFT; HOWARD, 1981). In this paper, the authors showed that the axiom of choice is equivalent to the representation theorem for multialgebras. Before Grätzer, Bjarni Jonsson and Alfred Tarski, in (JONSSON; TARSKI, 1951, p. 933), also introduced a representation theorem for complex algebras (defined briefly in Section 1.3). But, we present here the representation theorem of Grätzer instead of the theorem of Jonsson and Tarski given that the focus of the work are the multialgebras seen as algebras with multioperations (following the Marty’s line), instead of Jonsson and Tarski perspective.

1.4 Hyperrings and hyperfields. In this section, we will briefly talk about the origin and definition of other hyperstructures, namely as hyperrings and hyperfields.


42

A hyperring is a generalization of a ring where one of the operations is a hyperoperation and similarly, hyperfield is the structure that generalizes the usual concept of field. The concept of hyperfield was introduced by Marc Krasner in (KRASNER, 1957) in connection with his work on valued fields. Definition 1.4.1. (KRASNER, 1957; KRASNER, 1983) A hyperfield hC, +, ·i is a set C with a operation (·) : C × C → C and with a multioperation (+) : C × C → P(C). According to the established, the use of this multioperation can be extended S to subsets of C as following: A + B = (a + b), for a ∈ A, b ∈ B, a + B = {a} + B and A + b = A + {b}. The structure hC, +, ·i satisfies the following properties: (i) Properties of the operation (·): a) C is a multiplicative semigroup20 with respect to operation (·) and has a bilaterally absorbing element21 denoted by 0; b) C − {0} is a group with respect to operation · and the identity element is denoted by 1. (ii) Properties of the multioperation (+): a) For every a, b ∈ C, a + b = b + a (commutativity) b) For every a, b, c ∈ C, (a + b) + c = a + (b + c) (associativity) c) For every a ∈ C, there exists one and only one a0 ∈ C, such that 0 ∈ a + a022 (inverse element) d) For every a, b, c ∈ C, if c ∈ a + b then b ∈ c + (−a) (quasi subtration) (iii) Properties of distributivity: a) For every a, b, c ∈ C, c · (a + b) = c · a + c · b b) For every a, b, c ∈ C, (a + b) · c = a · c + b · c The concept of hyperring was introduced in 1941, by Robert S. Pate in the paper intitled “Rings with multiple-valued operations” (PATE, 1941). The main difference between the hyperrings and the usual rings is that in a hyperring, the addition is not necessarily unique. 20

21

22

See the Chapter 0 and (RASIOWA; SIKORSKI, 1963) for more about definitions and properties of usual groups/fields. Let hS, ·i be a system composed by a set S and a binary operation ·, a bilaterally absorbing element on hS, ·i is the element z such that, for every s in S, z · s = s · z = z. a0 will be denoted by −a.


43

When modifying an axiom of the definition of Marc Krasner of hyperfield, we can also get a hyperring definition. So, according to Marc Krasner (KRASNER, 1957; KRASNER, 1983), we have that a hyperring is a structure hC, +, ·i composed by a set C, an operation (·) : C × C → C and a multioperation (+) : C × C → P(C). The structure hC, +, ·i satisfy the properties of the items (ii) and (iii) and the property (i) (of the Definition 1.4.1) is replaced by the following property: (i)’ C is a multiplicative semigroup with bilaterally absorbing element 0. Marc Krasner define also a subhyperring as a subset C 0 of a hyperring C such that C 0 is closed for the multioperation (+), for the operation (·) and for the inverse element, that is, if a, b ∈ C 0 then a + b ∈ C 0 and a · b ∈ C 0 and if a ∈ C 0 , then −a ∈ C 0 . In the next section, we will introduce the definition of other kind of hyperstructures, obtained by replacing some axioms (as the axiom of associativity and commutativity), by weaker versions.

1.5 Hv -structures The Hv -structure concept was introduced by Thomas Vougiouklis, in his paper “The fundamental relation in hyperrings. The general hyperfield” (VOUGIOUKLIS, 1991). In the quote below, Vougiouklis reveals his motivation for the introduction of this concept: This paper concludes with the definition of a new class of hyperstructures, more general than the known ones, introduced by the author at the Fourth International Congress on Algebraic Hyperstructures and Applications (AHA). The motivation to introduce this class was the uniting elements procedure [2]23 . (VOUGIOUKLIS, 1991, p. 210, our translation).

In the quote, the uniting elements procedure is a method that allows you to put in the same class two or more elements. Vougiouklis (VOUGIOUKLIS, 2014) claims that, by means of hyperstructures, this method leads to structures with additional properties. The Hv -structures are weaker generalizations of some hyperstructures, for example hypergroups, hyperrings, hyperlattices, and so on. In the Hv -structures, some axioms are replaced by corresponding weaker versions. See, for example, the definition of a Hv -group: Definition 1.5.1. (VOUGIOUKLIS, 1991) A Hv -group hG, ·i is a set G equipped with a multioperation (·) : G × G → (P(G) − {∅}) that satisfies the following axioms: 23

By [2] the author refers to his paper of 1989 written together with Piergiulio Corsini: “From groupoids to groups through hypergroups” (CORSINI; VOUGIOUKLIS, 1989).


44

1. For every a, b, c ∈ G, ((a · b) · c) ∩ (a · (b · c)) 6= ∅ [weak associativity] 2. For every a ∈ G, a · G = G · a = G. In the definition above, the non-empty intersection, called by him weak associativity, substitutes the equality of the usual associativity. In the same paper, Thomas Vougiouklis says that structures like the Hv -groups, Hv -rings and Hv -fields and so on, can be defined similarly, by replacing the associative law for its weaker version (1) and the commutative law by the following weaker version: (a · b) ∩ (b · a) 6= ∅ (for every a, b ∈ G). The definition of the Hv -groups motivated that several papers related to Hv -structures have appeared in the literature in recent years, as for example (VOUGIOUKLIS; SPARTALIS; KESSOGLIDES, 1997), (VOUGIOUKLIS, 1999), (DAVVAZ, 2000), (DAVVAZ, 2006), (VOUGIOUKLIS, 2008) and (VOUGIOUKLIS, 2014). With relation to the application of hyperstructures, a lot of work have been done in several areas of Mathematics (pure and applied) like in algebra, geometry, topology, graph theory, probability theory, theory of automata, fuzzy theory, and so on. However, this is not the only way to apply hyperstructures. In the next section, we will talk about other application of hyperstructures, strongly related to the non-determinism notion.

1.6 Non-deterministic matrices At the beginning of this chapter, we already highlight the strong relationship between non-deterministic matrices and hyperstructures. Now, we will talk about the origin of the concept of non-deterministic matrix. Non-deterministic matrix is a generalization of the usual concept of manyvalued matrix24 . So, before going to non-determinism, let us talk briefly about the origin of the algebraic theory of logical matrices. The idea of logical matrices was used by Pierce (PEIRCE, 1885) and by Schröder (SCHRÖDER, 1891). They applied truth-tables for manipulating logical problems, but both had as focus only the classical logic. According to (WÓJCICKI, 1984), the notion of logical matrix was defined in a more rigorous and general way by Łukasiewicz and Tarski in (ŁUKASIEWICZ; TARSKI, 1930) and the foundations of the theory of logical matrices were developed by Loś in (ŁOŚ, 1949) and in a series of papers of 24

Additional information about many-valued matrices can be found at (ROSSER; TURQUETTE, 1952), (BOLC; BOROWIK, 1992), (MALINOWSKI, 1993), (GOTTWALD, 2001) and (HAHNLE, 2001).


45

Kalicki (KALICKI, 1949), (KALICKI, 1950b), (KALICKI, 1950a) and (KALICKI, 1952). Wójcicki highlights the work of Wajsberg (WAJSBERG, 1935), Jaśkowski (JAŚKOWSKI, 1936), Tarski (TARSKI, 1938) and Suszko (SUSZKO, 1957) as being of considerable importance in the development of the theory of logical matrices. Formally: Definition 1.6.1. A logical matrix for a propositional language L(Σ, Ξ) can be defined as a pair M = hA, Di such that A = hA, σA i is an algebra25 over Σ with domain A, and D is a subset of A. The elements of D are called designated elements. Logical matrices are used to give a natural semantics for propositional logics and they also play an important role in the general techniques of algebraization of logics introduced by W. Blok and D. Pigozzi (BLOK; PIGOZZI, 1986; BLOK; PIGOZZI, 1989). However, the logical matrices can be used not only like semantics for bivalent logics (as in the case of classical propositional logic), but also for many-valued logics, as is the case of some modal logics and many-valued logics in general. Although several propositional logics can be characterized semantically using a many-valued matrix (ŁOŚ; SUSZKO, 1975), many of them have only infinite matrices and such matrices do not constitute a good decision procedure for these logics. So, an alternative solution is the use of the non-deterministic matrices. The aplication of non-deterministic matrices was quite studied by Avron and his colaborators. At this point we cannot speak too much about the origin of this concept, but we discovered that in 1962, N. Rescher (RESCHER, 1962) was using non-deterministic matrices with the name of quasi-truth-functional systems. This concept also were used by J. Kearns in 1981 (KEARNS, 1981) and by Y. Ivlev in 1988 (IVLEV, 1988). We believe that all these authors gave their non-deterministic matrix definition independently. The non-deterministic matrix was studied by Avron and Lev firstly in (AVRON; LEV, 2001). Although they were not the first to introduce this concept, this idea was quite disseminated through his papers. So, in the next lines, we will give more attention for Avron and Lev’s definitions. Avron and Zamansky (AVRON; ZAMANSKY, 2011) motivated by the conflict between the truth-functionality principle and the non truth-functional character of information present in the real world, use non-deterministic matrices to weaken this principle. See bellow, an example of application of non-determinism. This example was based on Example “linguistic ambiguity” in (AVRON; ZAMANSKY, 2011, p. 3). Example 1.6.2. In the natural language, the word “or” can have two meanings: an inclusive and other exclusive. For example: 25

Recall the Definition 0.2.11 of an algebra in the Chapter 0.


46

(1) My father is, right now, playing soccer in Brazil or in Japan. (2) I’m going to buy the pair of shoes light blue or the pair of shoes dark blue. In the item (1), the disjunction “or” is exclusive, because a person cannot be in two places at the same time, but in the item (2) the disjunction “or” is inclusive, because if I’m with a doubt about which pair of shoes to buy, I can buy the two pair of shoes. The problem associated with the use of “or” is because, in many cases, we can’t distinguish if the “or” in question is inclusive or if it is exclusive, however, even in these cases, we would like to be able to infer something from what was said, and this can be done through the non-deterministic matrices, see below: ∨ 1

1

{1,0}

1

0

{1}

0

1

{1}

0

0

{0}

Now, we present the formal definition of non-deterministic matrix. Definition 1.6.3. (AVRON; LEV, 2001, p. 536) A non-deterministic matrix (or in short Nmatrix) for a propositional language L(Σ, Ξ)26 is a ordered triple M = hV, D, Oi, such that: (i) V is a non-empty set of truth-values, (ii) D is a proper and non-empty subset of V called set of designated truth values and (iii) O assigns one corresponding function c˜ : V n → (P(V ) − {∅}) for all n-ary connective c ∈ Σn . Note in the above definition that the function c˜ is a multioperation and so the non-deterministic matrices are multialgebras. One of the main features of the non-deterministic matrices is that the truth value of a complex formula can be chosen non-deterministically from a non-empty set of options. In (AVRON; LEV, 2001) we also found a definition of valuation and of semantics consequence in the non-deterministic matrices theory. These concepts will be used in the next chapters and, therefore, they will be presented now. 26

Recall the definition of L(Σ, Ξ) in the Chapter 0. Note that Avron denotes, in a simplified way, by L.


47

Definition 1.6.4. (AVRON; LEV, 2001, p. 536) A valuation in an Nmatrix M = hV, D, Oi is a function v : L(Σ, Ξ) → V such that, for every c ∈ Σn , α1 , ..., αn ∈ L(Σ, Ξ) and n ∈ N the following condition is satisfied: v(c(α1 , ..., αn )) ∈ c˜(v(α1 ), ..., v(αn )). Definition 1.6.5. (AVRON; LEV, 2001, p. 536) Let be ∆ ∪ {α} ⊆ L(Σ, Ξ), then ∆ M α if, for all valuation v in an Nmatrix M = hV, D, Oi, v[∆] ⊆ D ⇒ v(α) ∈ D. In particular, if ∆ = ∅, then M α if, for all valuation v in M, v(α) ∈ D. The non-deterministic matrices motivated the development of this work, because, even though much has already been done by Avron and his collaborators, the nondeterministic matrices have not been linked to multialgebras from the point of view of universal algebra. So, the next chapter is dedicated to presenting a non-deterministic algebraic theory based on multialgebras.

48

2 Some concepts in universal multialgebra

As has been said in the first chapter, multialgebras have been very much studied in the literature, but the generalization from algebra universal to multialgebras of even basic conceps such as homomorphism, subalgebras and congruences is far from obvious, and several different alternatives were proposed in the literature. In this chapter we begin the study of formal properties of multialgebras, with emphasis on universal algebra, through the development of complementary results to what has already been developed in the theory of hyperstructures and already mentioned in the Chapter 1 of this thesis. This study will enable the appropriate choice of concepts in multialgebra theory in order to obtain the results in subsequent chapters. The definitions and results presented in this chapter were adapted to our purpose in accordance with results from literature of multialgebras. The reference for this chapter is the Chapter 1, whereas it is clear that the definitions have been adapted to our context. Also worth to mention that the same definitions (or results) have been proposed by several authors using a different languages that one used in this Thesis. Therefore, it is difficult to adopt and to reference definitions (or results) from a specific author.

2.1 Multialgebras and homomorphisms In this section, we present several concepts in multialgebras theory that we will adopt along this work as well as results in the category of multialgebras. At some places these algebras can be called non-deterministic algebras 1 . Definition 2.1.1 (Multialgebra). Let Θ be a signature. A multialgebra (or hyperalgebra) over Θ is a pair A = hA, σA i such that A is a nonempty set (the universe or support of A) and σA is a mapping assigning to each c ∈ Θn , a function (called multioperation or hyperoperation) cA : An → (P(A) − {∅}). In particular, ∅ = 6 cA ⊆ A if c ∈ Θ0 . Notation 2.1.2. If c ∈ Θ0 such that cA is a singleton, then we will denote by cA the single element of cA ; that is cA = {cA }. On the other hand, the support of A will be frequently denoted by |A|. 1

Because of the Nmatrices of Avron and his colaborators (AVRON; LEV, 2001; AVRON; LEV, 2005; AVRON; KONIKOWSKA, 2005).

Chapter 2. Some concepts in universal multialgebra

49

Now, we can present a similar definition of non-deterministic matrix (Definition 1.6.3) using the concept of multialgebras: Definition 2.1.3 (Non-deterministic matrix). Let Θ a signature. A non-deterministic matrix (or Nmatrix) is a pair M = hA, Di such that A = hA, σA i is a multialgebra over Θ with support A, and D is a subset of A. The elements in D are called designated elements. From the definition of non-deterministic matrices, recall that the semantics2 associated to non-deterministic matrices is given by: Definition 2.1.4 (Valuation). Let M = hA, Di be a non-deterministic matrix over a signature Θ. A valuation3 over M is a function v : F or(Θ) → |A| such that, for every c ∈ Θn and every α1 , . . . , αn ∈ F or(Θ): v(c(α1 , . . . , αn )) ∈ cA (v(α1 ), . . . , v(αn )). In particular, v(c) ∈ cA , for every c ∈ Θ0 . Definition 2.1.5 (Consequence relation). Let M = hA, Di be a non-deterministic matrix over a signature Θ, and let Γ ∪ {α} ⊆ F or(Θ). We say that α is a consequence of Γ in the non-deterministic matrix M, denoted by Γ |=M α, if the following holds: for every valuation v over M, if v[Γ] ⊆ D then v(α) ∈ D. In particular, α is valid in M, denoted by |=M α, if v(α) ∈ D for every valuation v over M. Avron in (AVRON, 2005, p. 156 and p. 157) presents two non-deterministic B matrices MB 5 and M3 , that semantically characterize the logical system B, which is known in literature as mbC, one of the simplest Logics of Formal Inconsistency (LFIs) 4 . These non-deterministic matrices will be presented in the following two examples, and subsequently analyzed in the light of the concepts introduced, along with other nondeterministic matrices introduced in the literature. Example 2.1.6. Let Σ = {∧, ∨, →, ¬, ◦} be a signature and M5 = hA5 , D5 i be the non-deterministic matrix over Σ such that: (i) |A5 | = A5 = {t, tI , I, f, fI }; (ii) D5 = {t, tI , I}; 2

3

4

These definitions were adapted from the definitions of Avron (AVRON; LEV, 2001; AVRON; LEV, 2005; AVRON; KONIKOWSKA, 2005). It is worth noting that Avron and his collaborators (AVRON; LEV, 2001; AVRON, 2005; AVRON; ZAMANSKY, 2011) use the term legal valuation to refer to valuations over an Nmatrix. Introduced by W. Carnielli and J. Marcos in (CARNIELLI; MARCOS, 2002) and after studied in detail in (CARNIELLI; CONIGLIO; MARCOS, 2007) and (CARNIELLI; CONIGLIO, 2016).


50

(iii) For each connective c, the multioperation σA5 (c) = cA5 is defined by the following tables (here, F5 = {f, fI }). ∨A5

t

tI

I

f

fI

∧A5

t

tI

I

f

fI

t

D5

D5

D5

D5

D5

t

D5

D5

D5

F5

F5

tI

D5

D5

D5

D5

D5

tI

D5

D5

D5

F5

F5

I

D5

D5

D5

D5

D5

I

D5

D5

D5

F5

F5

f

D5

D5

D5

F5

F5

f

F5

F5

F5

F5

F5

fI

D5

D5

D5

F5

F5

fI

F5

F5

F5

F5

F5

→A5

t

tI

I

f

fI

t

D5

D5

D5

F5

F5

t

F5

t

D5

tI

D5

D5

D5

F5

F5

tI

F5

tI

F5

I

D5

D5

D5

F5

F5

I

D5

I

F5

f

D5

D5

D5

D5

D5

f

D5

f

D5

fI

D5

D5

D5

D5

D5

fI

D5

fI

F5

¬A5

◦A5

Clearly, M5 induces a multialgebra A5 = hA5 , σA5 i over Σ. Example 2.1.7. Let Σ = {∧, ∨, →, ¬, ◦} be a signature and let M3 = hA3 , D3 i be the non-deterministic matrix over Σ such that: (i) |A3 | = A3 = {t0 , I 0 , f 0 }; (ii) D3 = {t0 , I 0 }; (iii) For each connective c, the multioperation σA3 (c) = cA3 is defined by the following tables: ∨A3

t0

I0

f0

∧A3

t0

I0

f0

t0

D3

D3

D3

t0

D3

D3

{f 0 }

I0

D3

D3

D3

I0

D3

D3

{f 0 }

f0

D3

D3

{f 0 }

f0

{f 0 }

{f 0 }

{f 0 }

→A3

t0

I0

f0

t0

D3

D3

{f 0 }

0

D3

D3

f0

D3

D3

I

0

¬A3 t0

{f 0 }

◦A3 t0

{f }

I

0

D3

I

D3

f0

D3

f0

Let A3 = hA3 , σA3 i. So A3 is a multialgebra over Σ.

0

A3 {f 0 } A3


51

Remark 2.1.8. The example below will be denoted by M0 3 although it is a 5-valued matrix because M0 3 is an extension of the 3-valued matrix M3 . Example 2.1.9. Let Σ = {∧, ∨, →, ¬, ◦} be a signature and M0 3 = hA0 3 , D30 i be the non-deterministic matrix over Σ such that: (i) |A0 3 | = A03 = {t0 , t0I , I 0 , f 0 , fI0 }; (ii) D30 = {t0 , I 0 }; 0

(iii) For each connective c, the multioperation σA0 3 (c) = cA 3 is defined by the following tables (here, F30 = {f 0 }). 0

0

∨A 3

t0

t0I

I0

f0

fI0

∧A 3

t0

t0I

I0

f0

fI0

t0

D30

D30

D30

D30

D30

t0

D30

D30

D30

F30

F30

t0I

D30

D30

D30

D30

D30

t0I

D30

D30

D30

F30

F30

I0

D30

D30

D30

D30

D30

I0

D30

D30

D30

F30

F30

f0

D30

D30

D30

F30

F30

f0

F30

F30

F30

F30

F30

fI0

D30

D30

D30

F30

F30

fI0

F30

F30

F30

F30

F30

0

0

0

→A 3

t0

t0I

I0

f0

fI0

¬A 3

◦A 3

t0

D30

D30

D30

F30

F30

t0

F30

t0

{t0 , I 0 , f 0 }

t0I

D30

D30

D30

F30

F30

t0I

F30

t0I

{t0 , I 0 , f 0 }

I0

D30

D30

D30

F30

F30

I0

D30

I0

F30

f0

D30

D30

D30

D30

D30

f0

D30

f0

{t0 , I 0 , f 0 }

fI0

D30

D30

D30

D30

D30

fI0

D30

fI0

{t0 , I 0 , f 0 }

Clearly, M0 3 induces a multialgebra A0 3 = hA03 , σA0 3 i over Σ. Definition 2.1.10 (Homomorphisms of multialgebras). Let A = hA, σA i and B = hB, σB i be two multialgebras over a signature Θ, and let h : A → B be a function. (i) h is said to be a homomorphism5 from A to B, denoted by h : A → B, if h[cA (~a)] ⊆ cB (h(a1 ), . . . , h(an )) for every c ∈ Θn and ~a ∈ An . In particular, h[cA ] ⊆ cB for every c ∈ Θ0 . (ii) h is said to be a full homomorphism6 from A to B, which is denoted by h : A →s B, if h[cA (~a)] = cB (h(a1 ), . . . , h(an )) for every c ∈ Θn and ~a ∈ An . In particular, h[cA ] = cB for every c ∈ Θ0 . 5 6

Nolan called weak homomorphism, see Definition 1.3.7. Nolan called strong homomorphism, see Definition 1.3.7.


52

Notation 2.1.11. When there is no risk of confusion, a full homomorphism will be simply denoted by h : A → B instead of h : A →s B. Example 2.1.12. Let A5 = hA5 , σA5 i and A3 = hA3 , σA3 i be the multialgebras introduced in the Examples 2.1.6 and 2.1.7. Let h : A5 → A3 be a function such that h(t) = h(tI ) = t0 , h(I) = I 0 and h(f ) = h(fI ) = f 0 . Clearly, h[D5 ] = D3

and

h(F5 ) = {f 0 }.

From here, and by definitions of σA5 and σA3 , is immediate that h[cA5 (a, b)] ⊆ cA3 (h(a), h(b)) for every a, b ∈ A5 and c ∈ {∨, ∧, →} (in fact, the equality holds). In the same way h[¬A5 a] ⊆ ¬A3 h(a) for every a ∈ A5 (again, the equality holds). Finally, h[◦A5 a] = h[D5 ] = D3 ⊂ A3 = ◦A3 h(a), for a ∈ {t, f }, while h[◦A5 I] = h[F5 ] = {f 0 } = ◦A3 I 0 = ◦A3 h(I) and h[◦A5 a] = h[F5 ] = {f 0 } ⊂ A3 = ◦A3 h(a), for a ∈ {tI , fI }. For example, h[◦A5 t] = h[D5 ] = D3 ⊂ A3 = ◦A3 h(t) = ◦A3 t0 , and h[◦A5 fI ] = h[F5 ] = {f 0 } ⊂ A3 = ◦A3 f 0 = ◦A3 h(fI ). So, h[cA5 a] ⊆ cA3 h(a) for every a ∈ A5 and c ∈ {¬, ◦} and therefore h defines a homomorphism h : A5 → A3 . Example 2.1.13. Let A5 = hA5 , σA5 i and A3 = hA3 , σA3 i be the multialgebras introduced in the Examples 2.1.6 and 2.1.7. Let h : A3 → A5 the function such that h(t0 ) = I, h(I 0 ) = fI and h(f 0 ) = tI . We have that h[◦A3 I 0 ] = h[f 0 ] = {tI } * F5 = ◦A5 fI = ◦A5 h(I 0 ). It shows that there is b ∈ A3 , such that h[◦A3 b] * ◦A5 h(b) and so, h does not define a homomorphism h : A3 → A5 . However, we still note that h[◦A3 a] = h[A3 ] = {fI , tI , I} * F5 = ◦A5 h(a) for a ∈ {t0 , f 0 } that is h[◦A3 a] * ◦A5 h(a) for all a ∈ A3 . When there is no risk of confusion, we will assume that multialgebras are defined over a fixed signature Θ. Just as Benado (BENADO, 1955) introduced the concept of submultilattice, other authors such as Pickett (PICKETT, 1967), Hansoul (HANSOUL, 1981) and Pelea (PELEA, 2003) also studied the concepts of submultialgebra. In the next section, we will introduced some new results related to submutialgebras.

2.2 Submultialgebras From the definitions of homomorphisms, we will now analyze the notion of submultialgebra.


53

Definition 2.2.1 (Submultialgebra). Let B = hB, σB i and A = hA, σA i be two multialgebras over Θ. Then B is said to be a submultialgebra of A, denoted by B ⊆ A, if the following conditions hold: (i) B ⊆ A, (ii) if c ∈ Θn and ~b ∈ B n , then cB (~b) ⊆ cA (~b); in particular, cB ⊆ cA if c ∈ Θ0 . In Proposition 2.5.1 the meaning of submultialgebras will be clarified. Example 2.2.2. Let A3 = hA3 , σA3 i and A0 3 = hA03 , σA0 3 i be the multialgebras introduced in the Examples 2.1.7 and 2.1.9. We can see that A3 ⊆ A03 and by definitions of σA3 and σA0 3 we have, for every c ∈ Θn and ~a ∈ An3 , 0 cA3 (~a) ⊆ cA 3 (~a). Therefore, A3 is submultialgebra of A0 3 , that is, A3 ⊆ A0 3 . 0

In fact, we have more, we have cA3 (~a) = cA 3 (~a). In a similar way to universal algebra, we can apply the concept of subuniverse to multialgebras. Definition 2.2.3 (Submultiuniverse). Let A = hA, σA i a multialgebra. A submultiuniverse of A is a non-empty subset B of A which is closed under the multioperations of A, that is, for every n ≥ 0, c ∈ Θn and ~b ∈ B n , cA (~b) ⊆ B. Note that if B = hB, σB i is a submultialgebra of A = hA, σA i, then B is a submultiuniverse of A. So: Example 2.2.4. Let A3 = hA3 , σA3 i and A0 3 = hA03 , σA0 3 i be the multialgebras introduced in the Examples 2.1.7 and 2.1.9. We have that A3 is submultiuniverse of A0 3 . Definition 2.2.5 (Generated submultiuniverse). Let A = hA, σA i be a multialgebra and ∅= 6 X ⊆ A. The submultiuniverse of A generated by X, denoted by sg A (X) (in short sg(X)) is defined by: sg(X) =

\

{B : B is submultiuniverse of A and X ⊆ B}.

Remark 2.2.6. Note that A is submultiuniverse of A that contains X. So, sg(X) is well defined, since the set {B : B is submultiuniverse of A and X ⊆ B} is non-empty. Proposition 2.2.7. If A = hA, σA i is a multialgebra and ∅ 6= X ⊆ A, then sg(X) is, in fact, a submultiuniverse of A.


54

Proof. Note that sg(X) ⊆ A and sg(X) 6= ∅. Let n ≥ 0, c ∈ Θn and b1 , . . . , bn ∈ sg(X). If B is a subuniverse of A such that X ⊆ B, then cA (b1 , . . . , bn ) ⊆ B, thus cA (b1 , . . . , bn ) ⊆ sg(X) and so sg(X) is a submultiuniverse of A. As in the case of usual algebras, we can give a constructive definition of the set sg(X). Proposition 2.2.8. Let A = hA, σA i be a multialgebra, ∅ 6= X ⊆ A and let {E n (X) : n ≥ 0} be a family of subsets of A, defined by recursion as follows: E 0 (X) = X E n+1 (X) = E n (X) ∪

S

{cA (a1 , . . . , ak ) : k ≥ 0, c ∈ Θk and a1 , . . . , ak ∈ E n (X)}.

Then, sg(X) = {E n (X) : n ≥ 0}. S

Proof. In (AMERI; ROSENBERG, 2009, Theorem 3.17, p. 11).

Lemma 2.2.9. Let A = hA, σA i and B = hB, σB i be two multialgebras, ∅ 6= X ⊆ A and let h : A → B be a homomorphism. If E n (X) and E n (h[X]) are defined recursively as in Proposition 2.2.8 then h[E n (X)] ⊆ E n (h[X]). Proof. The proof is by induction over n, n ≥ 0. If n = 0, h[E 0 (X)] = h[X] = E 0 (h[X]). Suppose that h[E n (X)] ⊆ E n (h[X]); then h[E n+1 (X)] = h[E n (X) ∪ {cA (a1 , . . . , ak ) : k ≥ 0, c ∈ Θk and a1 , . . . , ak ∈ E n (X)}] = S h[E n (X)] ∪ h[ {cA (a1 , . . . , ak ) : k ≥ 0, c ∈ Θk and a1 , . . . , ak ∈ E n (X)}] ⊆ S E n (h[X]) ∪ {h[cA (a1 , . . . , ak )] : k ≥ 0, c ∈ Θk and a1 , . . . , ak ∈ E n (X)}) ⊆ S E n (h[X]) ∪ {cB (h(a1 ), . . . , h(ak )) : k ≥ 0, c ∈ Θk and h(a1 ), . . . , h(ak ) ∈ E n (h(X))} = E n+1 (h[X]). S

Remark 2.2.10. In the previous lemma, if h is a full homomorphism, then h[E n (X)] = E n (h[X]). Theorem 2.2.11. Let A = hA, σA i and B = hB, σB i be two multialgebras, ∅ = 6 X ⊆A and let h : A → B be a homomorphism. Then, h[sg(X)] ⊆ sg(h[X]).


55

Proof. By Proposition 2.2.8, we have h[

h[sg(X)] = h

i

{E n (X) : n ≥ 0} =

{h[E n (X)] : n ≥ 0}.

[

Using the previous lemma and Proposition 2.2.8, we have [

{h[E n (X)] : n ≥ 0} ⊆

[

{E n (h[X]) : n ≥ 0} = sg(h[X]).

Definition 2.2.12 (Direct image). Let A = hA, σA i and B = hB, σB i be two multialgebras, and let h : A → B be a homomorphism. The direct image of h is the multialgebra Sn h(A) = hh[A], σh(A) i, such that, for every c ∈ Θn and ~b ∈ h[A], ch(A) (~b) = h[cA (~a)] : o ~a ∈ h−1 (~b) . In particular, ch(A) = h[cA ] for every c ∈ Θ0 . Proposition 2.2.13. Let A = hA, σA i and B = hB, σB i be two multialgebras, and let h : A → B be a homomorphism. The direct image of h, h(A) = hh[A], σh(A) i is submultialgebra of B. Proof. If ~b ∈ h[A] and ~a ∈ h−1 (~b) then h[cA (~a)] ⊆ cB (h(~a)) = cB (~b). Hence ch(A) (~b) ⊆ cB (~b). So, h(A) ⊆ B. Although some of the results that were presented, already exist in the literature of multialgebras, the notion of interpretation of formulas has not yet been properly studied and it will be the topic of the next section.

2.3 Interpretation of formulas In the book “Fuzzy algebraic hyperstructures: an introduction”, the authors defined a semihypergroup from the definition of hypergroupoid using a generalized associativity law, that is: Definition 2.3.1. (DAVVAZ; CRISTEA, 2015, Definition 1.3.2, p. 17) A semihypergroup is a hypergroupoid hS, ·i such that for all a, b, c ∈ S, [ u∈a·b

u·c=

[

a·v

v∈b·c

From generalized associativity law and by definition of legal valuation (Definition 2.1.4), we can think about define a notion of interpretation of formulas in multialgebras theory. For instance, suppose that if c ∈ Θ0 , then cA is a singleton7 (that is, cA = {cA }) and consider the two definitions below: 7

Remember of the Notation 2.1.2.


56

Definition 2.3.2 (Assignment). Let A = hA, σA i be a multialgebra over a signature Θ and let Ξ be the set of variables. An assignment in A is a function ρ : Ξ → A. Definition 2.3.3 (Interpretation of formulas). Let A = hA, σA i be a multialgebra and let ρ be an assignment in A. The multifunction (·)Aρ : L(Θ, Ξ) →M A is an interpretation αAρ of the formula α in A by ρ, and it is defined, by induction on the complexity of the formula α, as being the non-empty subset of A such that: (i) ξ Aρ = {ρ(ξ)}, if ξ ∈ Ξ; (ii) cAρ = {cA }, if c ∈ Θ0 ; (iii) c(α1 , . . . , αn )Aρ = {cA (~a) : ai ∈ αiAρ , for 1 ≤ i ≤ n}, if n > 0, c ∈ Θn and αi ∈ L(Θ, Ξ) (for 1 ≤ i ≤ n). S

After looking quickly through the example below we can think that the last is a good definition of interpretation of formulas in a multialgebra: Example 2.3.4. Let A3 = hA3 , σA3 i be the multialgebra introduced in the Example 2.1.7 and let ρ be an assignment in A3 . The interpretation of the formula ¬ ◦ ξ → (ξ ∧ ¬ξ) in A3 , for ρ(ξ) = t0 is obtained as follows: (¬ ◦ ξ → (ξ ∧ ¬ξ))A3 ρ = {→A3 (a1 , a2 ) : a1 ∈ (¬ ◦ ξ)A3 ρ and a2 ∈ (ξ ∧ ¬ξ)A3 ρ }. But, S S (¬ ◦ ξ)A3 ρ = {¬A3 (a) : a ∈ (◦ξ)A3 ρ } and (◦ξ)A3 ρ = {◦A3 (a) : a ∈ ξ A3 ρ }. On the other S S hand (ξ∧¬ξ)A3 ρ = {∧A3 (a1 , a2 ) : a1 ∈ ξ A3 ρ and a2 ∈ (¬ξ)A3 ρ }. But (¬ξ)A3 ρ = {¬A3 (a) : a ∈ ξ A3 ρ } and ξ A3 ρ = {ρ(ξ)}. Since ρ(ξ) = t0 we have (¬ξ)A3 ρ = ¬A3 (t0 ) = {f 0 }, (◦ξ)A3 ρ = ◦A3 (t0 ) = {t0 , I 0 , f 0 }. So, (¬ ◦ ξ)A3 ρ = {t0 , I 0 , f 0 }. Thus, (ξ ∧ ¬ξ)A3 ρ = ∧A3 (t0 , f 0 ) = {f 0 }. Therefore (¬ ◦ ξ → (ξ ∧ ¬ξ))A3 ρ = {f 0 } ∪ {f 0 } ∪ {t0 , I 0 , f 0 } = {t0 , I 0 , f 0 }. S

But see in the following example that this notion of interpretation does not always works well: Example 2.3.5. Let A5 = hA5 , σA5 i be the multialgebra introduced in the Example 2.1.6, let α the formula schema given by α = ¬(ξ1 ∨ ξ2 ) and let ρ be an assignment in A5 such that ρ(ξ1 ) = {I} and ρ(ξ2 ) = {f }. Then, αA5 ρ = A5 and (¬α)A5 ρ = A5 and thus, (α ∨ ¬α)A5 ρ = A5 . But, since this formula schema is an axiom of mbC, its interpretation should belong to D5 . So, the question of defining the notion of interpretation of formulas in multialgebras is still an open problem.


57

We already mentioned in Chapter 1 that the congruence concept was defined in the environment of multialgebras by Schweigert (SCHWEIGERT, 1985) and it was extensively studied by Reza Ameri and Ivo G. Rosenberg (AMERI; ROSENBERG, 2009). In the next section, we will present this concept.

2.4 Multicongruences and quotient multialgebras The concepts of congruence and quotient algebra are fundamental tools in the algebraization theory. Thus the generalization of these concepts to multialgebras theory will be fundamental tools to obtain the results in the next chapters with respect to algebraization of logical systems. The natural generalization of the notion of congruence to multialgebras is as follows: Definition 2.4.1 (Multicongruence). Let A = hA, σA i be a multialgebra, and let θ ⊆ A×A. Then θ is said to be a multicongruence over A if the following properties hold: (i) θ is an equivalence relation; (ii) for every n > 0, c ∈ Θn and ~a, ~b ∈ An : if (ai , bi ) ∈ θ for every 1 ≤ i ≤ n then, for every a ∈ cA (~a) there is b ∈ cA (~b) such that (a, b) ∈ θ; (ii) for every c ∈ Θ0 and every a, b ∈ A: if a, b ∈ cA then (a, b) ∈ θ. Remark 2.4.2. Observe that, since θ is an equivalence relation, from Definition 2.4.1 (ii), we can obtain the symmetric condition: (ii’) for every n > 0, c ∈ Θn and ~a, ~b ∈ An : if (ai , bi ) ∈ θ for every 1 ≤ i ≤ n then, for every b ∈ cA (~b) there is a ∈ cA (~a) such that (a, b) ∈ θ. Example 2.4.3. Let A0 3 be the multialgebra introduced in the Example 2.1.9 and θ = {(t0I , t0 ), (t0 , t0I ), (fI0 , f 0 ), (f 0 , fI0 )} ∪ {(a, a) : a ∈ A03 } ⊆ A0 3 × A0 3 . It’s easy to see that θ is an equivalence relation. 0

Let a1 , b1 ∈ A03 and c ∈ {¬, ◦}, if (a1 , b1 ) ∈ θ then, for every a ∈ cA 3 (a1 ), there 0 is b ∈ cA 3 (b1 ) such that (a, b) ∈ θ. See, for example, if c = ◦:


58 0

0

(a1 , b1 ) ∈ θ

◦A 3 (a1 )

◦A 3 (b1 )

(t0 , t0I )

{t0 , I 0 , f 0 }

{t0 , I 0 , f 0 }

(t0I , t0 )

{t0 , I 0 , f 0 }

{t0 , I 0 , f 0 }

(f 0 , fI0 )

{t0 , I 0 , f 0 }

{t0 , I 0 , f 0 }

(fI0 , f 0 )

{t0 , I 0 , f 0 }

{t0 , I 0 , f 0 }

(I 0 , I 0 )

{f 0 }

{f 0 }

(a0 , a0 ) : a0 ∈ A03 − {I 0 }

{t0 , I 0 , f 0 }

{t0 , I 0 , f 0 } 0

0

Note that, in all lines of the table we have ◦A 3 (a1 ) = ◦A 3 (b1 ). We also have that if (a1 , b1 ) ∈ θ and (a2 , b2 ) ∈ θ, for every a1 , b1 , a2 , b2 ∈ A03 and 0 0 c ∈ {∧, ∨, →} then, for every a ∈ cA 3 (a1 , a2 ) there is b ∈ cA 3 (b1 , b2 ) such that (a, b) ∈ θ. See, for example, if c = ∨:

(a1 , b1 ) ∈ θ

(t0 , t0I )

(t0I , t0 )

(f 0 , fI0 )

(fI0 , f 0 )

(a0 , a0 ) : a ∈ {t0 , t0I , I 0 }

(a0 , a0 ) : a ∈ {f 0 , fI0 }

(t0 , t0I )

D30

D30

D30

D30

D30

D30

(t0I , t0 )

D30

D30

D30

D30

D30

D30

(f 0 , fI0 )

D30

D30

F0

F0

D30

F0

(fI0 , f 0 )

D30

D30

F0

F0

D30

F0

(a0 , a0 ) : a ∈ {t0 , t0I , I 0 }

D30

D30

D30

D30

D30

D30

(a0 , a0 ) : a ∈ {f 0 , fI0 }

D30

D30

F0

F0

D30

F0

0

0

Note that, in all lines of the table we have a ∈ cA 3 (a1 , a2 ) = cA 3 (b1 , b2 ). Therefore, θ is a multicongruence over A0 3 . Example 2.4.4. Let A0 3 be the multialgebra introduced in the Example 2.1.9 and θ0 = {(t0I , I 0 ), (I 0 , t0I ), (fI0 , f 0 ), (f 0 , fI0 )} ∪ {(a, a) : a ∈ A03 } ⊆ A0 3 × A0 3 . It’s easy to see that θ0 is an equivalence relation. 0

0

If c = ¬, since (I 0 , t0I ) ∈ θ0 , then ¬A 3 (I 0 ) = {t0 , I 0 }. But, ¬A 3 (t0I ) = {f 0 } and is not the case that (t0 , f 0 ) ∈ θ0 . So, there isn’t b ∈ {f 0 }, such that, for every a ∈ {t0 , I 0 }, (a, b) ∈ θ0 . Therefore, θ0 is not a multicongruence over A0 3 . Definition 2.4.5 (Quotient multialgebra). Let A = hA, σA i be a multialgebra, and let θ be a multicongruence over A. The quotient multialgebra (or factor multialgebra) of A modulo θ is the multialgebra A/θ = hA/θ , σA/θ i such that, for every c ∈ Θn and n o every (a1 /θ , . . . , an /θ ) ∈ (A/θ )n , cA/θ (a1 /θ , . . . , an /θ ) = a/θ : a ∈ cA (~a) . In particular, n

o

cA/θ = a/θ : a ∈ cA for every c ∈ Θ0 .


59

Proposition 2.4.6. If A = hA, σA i is a multialgebra and θ is a multicongruence over A, then A/θ = hA/θ , σA/θ i (as in previous definition) is, in fact, a multialgebra with the same type. Proof. Let θ be a multicongruence such that (ai , bi ) ∈ θ (for every 1 ≤ i ≤ n) and let c ∈ Θn (for n > 0). If a/θ ∈ {a/θ : a ∈ cA (~a)} then, since a ∈ cA (~a), there is b ∈ cA (~b) such that (a, b) ∈ θ. That is, a/θ = b/θ and a/θ ∈ {b/θ : b ∈ cA (~b)}. Similarly, we can show that {b/θ : b ∈ cA (~b)} ⊆ {a/θ : a ∈ cA (~a)}. Definition 2.4.7 (Canonical map). Let A = hA, σA i be a multialgebra, and let θ be a multicongruence over A. The canonical map p : A → A/θ is given by p(a) = a/θ for every a ∈ A. Proposition 2.4.8. Let A be a multialgebra and let θ be a multicongruence over A. Then the canonical map p : A → A/θ is a (full) homomorphism of multialgebras such that p(A) = A/θ . Proof. p[cA (a1 , . . . , an )] = cA (a1 , . . . , an )/θ = cA/θ (a1 /θ, . . . , an /θ) = cA/θ (p(a1 ), . . . , p(an )). In particular, if c ∈ Θ0 , then p[cA ] = cA /θ = cA/θ . Definition 2.4.9 (Compatible homomorphism). Let A = hA, σA i and B = hB, σB i be two multialgebras, let θ be a multicongruence over A and let h : A → B be a homomorphism. We say that h is a homomorphism compatible with θ, if for every a, b ∈ A, such that (a, b) ∈ θ, then h(a) = h(b). Theorem 2.4.10 (Homomorphism theorem). Let A = hA, σA i and B = hB, σB i be two multialgebras, let θ be a multicongruence over A, let h : A → B be a homomorphism compatible with θ and let p : A → A/θ be the homomorphism of the Proposition 2.4.8. ¯ from A/θ to B, such that h(a/θ) ¯ Then, there is a unique homomorphism h = h(a), that is ¯ ◦ p = h. h ¯ is well defined. Proof. For every a, b ∈ A, if (a, b) ∈ θ then h(a) = h(b). Thus, h ¯ A/θ (a1 /θ, . . . , an /θ)] = h[c ¯ A (a1 , . . . , an )/θ] = h[cA (a1 , . . . , an )] ⊆ h[c ¯ 1 /θ), . . . , h(a ¯ n /θ)). cB (h(a1 ), . . . , h(an )) = cB (h(a ¯ is a homomorphism and if h1 is another homomorphism from A/θ to B such Therefore, h ¯ are the same. that h1 (a/θ) = h(a), then h1 (a/θ) = h(a) = h(a/θ). So, h1 and h Example 2.4.11. Let A0 3 be the multialgebra introduced in the Example 2.1.9 and let θ = {(t0I , t0 ), (t0 , t0I ), (fI0 , f 0 ), (f 0 , fI0 )} ∪ {(a, a) : a ∈ A03 } be the congruence over A0 3 introduced in the Example 2.4.3.


60

If A0 3 /θ = hA03 /θ, σA0 3 /θ i, such that A03 /θ = {t0 /θ, I 0 /θ, f 0 /θ}, D30 /θ = {t0 /θ, I 0 /θ} 0 and for every c ∈ Σ, the multioperation σA0 3 /θ (c) = cA 3 /θ is defined by the following tables: ∨A 3 /θ

0

t0 /θ

I 0 /θ

f 0 /θ

∧A 3 /θ

0

t0 /θ

I 0 /θ

f 0 /θ

t0 /θ

D3 /θ

D3 /θ

D3 /θ

t0 /θ

D3 /θ

D3 /θ

{f 0 /θ}

I 0 /θ

D3 /θ

D3 /θ

D3 /θ

I 0 /θ

D3 /θ

D3 /θ

{f 0 /θ}

f 0 /θ

D3 /θ

D3 /θ

{f 0 /θ}

f 0 /θ

{f 0 /θ}

{f 0 /θ}

{f 0 /θ}

0

0

0

→A 3 /θ

t0 /θ

I 0 /θ

f 0 /θ

¬A 3 /θ

◦A 3 /θ

t0 /θ

D3 /θ

D3 /θ

{f 0 /θ}

t0 /θ

{f 0 /θ}

t0 /θ

A03 /θ

I 0 /θ

D3 /θ

D3 /θ

{f 0 /θ}

I 0 /θ

D3 /θ

I 0 /θ

{f 0 /θ}

f 0 /θ

D3 /θ

D3 /θ

D3 /θ

f 0 /θ

D3 /θ

f 0 /θ

A03 /θ

Then, A0 3 /θ is a quotient multialgebra of A0 3 modulo θ over Σ. Example 2.4.12. Let A0 3 /θ = hA03 /θ, σA0 3 /θ i and A3 = hA3 , σA3 i be the multialgebras introduced in the Examples 2.4.11 and 2.1.7, and let h1 : A03 /θ → A3 be the function such that h1 (t0 /θ) = t0 , h1 (I 0 /θ) = I 0 and h1 (f 0 /θ) = f 0 . So, h1 [D30 /θ] = D3

and

h1 [{f 0 /θ}] = {f 0 }.

And, therefore h1 define a (full) homomorphism h1 : A0 3 /θ → A3 .

2.5 The category of multialgebras Several authors, some of them already mentioned in the Chapter 1, studied the categories in which the objects are multialgebras. In particular, Nolan (NOLAN, 1979) defines five different homomorphisms between multialgebras and studies the category obtained with each one. Note that due to the large number of possibilities for each definition in multialgebra theory, the study of categories on multialgebras does not always coincide. In this section we will introduce the basic notions and results concerning the category of multialgebras that will be used mainly for what will be done in the Chapter 4. This results are based on the pre-print (CONIGLIO; ORELLANO; GOLZIO, 2016). Proposition 2.5.1. If B and A are two multialgebras over Θ such that |B| ⊆ |A| then: B ⊆ A iff the inclusion map i : |B| → |A| is a homomorphism from B to A. Proof. For every c ∈ Θn and ~b ∈ |B|n , (⇒) i[cB (~b)] = cB (~b) ⊆ cA (~b) = cA (i(b1 ), . . . , i(bn )). In particular, if c ∈ Θ0 , i[cB ] = cB ⊆ cA .


61

(⇐) cB (~b) = i(cB (~b)) ⊆ cA (i(b1 ), . . . , i(bn )) = cA (~b). In particular, if c ∈ Θ0 , cB = i[cB ] ⊆ cA . Proposition 2.5.2. Let Θ a signature. There is a category of multiagebras over Θ together with their (full) homomorphisms, that will be denoted by MAlg(Θ). Proof. Observe that, if f : A → B and g : B → C are homomorphisms of multialgebras, then g ◦ f : A → C is also a homomorphism of multialgebras: g ◦ f [cA (~a)] = g[f [cA (~a)]] ⊆ g[cB (f (a1 ), . . . , f (an ))] ⊆ cC (g(f (a1 )), . . . , g(f (an ))) = cC (g ◦ f (a1 ), . . . , g ◦ f (an )). The result still holds when we consider full homomorphisms. And, if f : A → B, g : B → C and h : C → D are homomorphisms of multialgebras, then because f, g and h are functions, clearly f ◦ (g ◦ h) = (f ◦ g) ◦ h. On the other hand, the identity mapping 1A : A → A is a (full) homomorphism from A to A, for every multialgebra A = hA, σA i: 1A [cA (~a)] = cA (~a) = cA (1A (a1 ), . . . , 1A (an )). In particular, 1A [cA ] ⊆ cA , if c ∈ Θ0 . And, if f 0 : B → A and g 0 : A → C are homomorphisms of multialgebras, then clearly 1A ◦ f 0 = f 0 and g 0 ◦ 1A = g 0 . The following results will be useful in the Chapter 4. Proposition 2.5.3. Let A = hA, σA i and B = hB, σB i be two multialgebras over Θ, and let f : A → B be a function. Then, f is an isomorphism f : A → B in the category MAlg(Θ) iff f is a full homomorphism f : A →s B which is a bijective function. Proof. (⇒) Since f : A → B is an isomorphism in the category MAlg(Θ), then f is a homomorphism and so f [cA (a1 , . . . , an )] ⊆ cB (f (a1 ), . . . , f (an )). On the other hand, since f is an isomorphism, then there is a homomorphism f −1 : B → A, such that f ◦ f −1 = 1B . So, cB (f (a1 ), . . . , f (an )) = f ◦ f −1 (cB (f (a1 ), . . . , f (an ))). Since, f −1 is a homomorphism, then f −1 [cB (f (a1 ), . . . , f (an ))] ⊆ cA (f −1 (f (a1 )), . . . , f −1 (f (an ))) and so, f (f −1 [cB (f (a1 ), . . . , f (an ))]) ⊆ f (cA (f −1 (f (a1 )), . . . , f −1 (f (an )))) = f [cA (a1 , . . . , an )] because f −1 ◦ f = 1A . Therefore, f is a full homomorphism. Suppose that f (a1 ) = f (a2 ) for some a1 , a2 ∈ A. Since f is an isomorphism, there is f −1 : B → A such that f −1 ◦ f = 1A , where 1A : A → A is the identity mapping. So, a1 = 1A (a1 ) = f −1 ◦ f (a1 ) = f −1 (f (a1 )) = f −1 (f (a2 )) = f −1 ◦ f (a2 ) = 1A (a2 ) = a2 . Therefore f is one-to-one. Let b ∈ B. Again, since f is an isomorphism, there is f −1 : B → A such that f ◦ f −1 = 1B , where 1B : B → B is the identity mapping. So, b = 1B (b) = f ◦ f −1 (b) = f (f −1 (b)) = f (a) for some a ∈ A and, therefore f is onto. (⇐) Suppose that f is a bijection; then, there is f −1 : B → A such that f −1 is the inverse function of f . Since f is a full homomorphism which is a surjection function, for every bi ∈ B with 1 ≤ i ≤ n, there is ai ∈ A, such that f (ai ) = bi , then f −1 [cB (~b)] =


62

f −1 [cB (f (a1 ), . . . , f (an ))] = f −1 [f ([cA (a1 , . . . , an )])] = f −1 ◦ f ([cA (a1 , . . . , an )]) = cA (~a) = cA (f −1 ◦ f (a1 ), . . . , f −1 ◦ f (an )) = cA (f −1 (b1 ), . . . , f −1 (bn )). Therefore, f −1 : B → A is a full homomorphism. So, for every a ∈ A, f −1 ◦ f (a) = f −1 (f (a)) = a = 1A (a) and for every b ∈ B, f ◦ f −1 (b) = f (f −1 (b)) = b = 1B (b). Therefore f is an isomorphism. Proposition 2.5.4. Let A = hA, σA i and B = hB, σB i be two multialgebras over Θ, and let f : A → B be a homomorphism. If f : A → B is an injective (one-to-one) function then f is a monomorphism in the category MAlg(Θ). Proof. Let f : A → B be a homomorphism in MAlg(Θ), such that f : A → B is an injection and let g, h : C → A be two homomorphisms in MAlg(Θ) such that f ◦ g = f ◦ h. If c ∈ |C|, then f (g(c)) = f ◦ g(c) = f ◦ h(c) = f (h(c)) and since f is an injection, g(c) = h(c). Therefore, f is a monomorphism. Proposition 2.5.5. Let A = hA, σA i and B = hB, σB i be two multialgebras over Θ, and let f : A → B be a function. Then, f is an epimorphism f : A → B in the category MAlg(Θ) iff f is a homomorphism in MAlg(Θ) such that f is a surjective (onto) function. Proof. (⇐) Let f : A → B be a surjective (onto) homomorphism in MAlg(Θ) and let g, h : B → C be two homomorphisms in MAlg(Θ) such that g ◦ f = h ◦ f . Since f is onto, for every b ∈ B, there is a ∈ A, such that f (a) = b. So, g(b) = g(f (a)) = g ◦ f (a) = h ◦ f (a) = h(f (a)) = h(b). Therefore g = h and so, f is an epimorphism. (⇒) Conversely, suppose that f : A → B is an epimorphism in MAlg(Θ) and let A0 be 0 a multialgebra over Θ with domain {0, 1} such that cA (~a) = {0, 1} for every c ∈ Θn 0 and ~a ∈ {0, 1}n . In particular, cA = {0, 1} for every c ∈ Θ0 . Consider g, h : B → {0, 1} such that g(x) = 1 if there exists y ∈ A such that x = f (y), g(x) = 0 otherwise and h(x) = 1 for every x ∈ B. Clearly, g and h are homomorphisms from B to A0 0 in MAlg(Θ): g[cB (~b)] ⊆ {0, 1} = cA (g(b1 ), . . . , g(bn )), for every c ∈ Θn and ~b ∈ B n . 0 In particular, if c ∈ Θ0 , then g[cB ] ⊆ {0, 1} = cA . The same is true for h. If a ∈ A, g ◦ f (a) = g(f (a)) = 1 = g(h(a)) = h ◦ f (a). So, g ◦ f = h ◦ f and since f is epimorphism in MAlg(Θ) then g = h. So, if b ∈ B, then g(b) = h(b) = 1 and by definition of g this means that f is a surjective (onto) function. In the next, we will present a definition of direct product in multialgebras. Definition 2.5.6 (Product in multialgebras). Let {Ai }i∈I be a family of multialgebras Q over Θ and let A = i∈I Ai be the standard cartesian product of the family of sets {Ai }i∈I Q with canonical projections πi : A → Ai for every i ∈ I. The (direct) product A = i∈I Ai of the family {Ai }i∈I is the multialgebra A = hA, σA i over Θ such that, for every c ∈ Θn Q Q and every ~a ∈ An , cA (~a) = i∈I cAi (πi (a1 ), . . . , πi (an )). In particular, cA = i∈I cAi for every c ∈ Θ0 .


63

In particular, the category MAlg(Θ) has a terminal object that corresponds to the product of the empty family of multialgebras over Θ. See the following proposition: Proposition 2.5.7 (Terminal object). The terminal object in MAlg(Θ) is the multialgebra 1 = h{∗}, σ1 i, such that c1 (∗, . . . , ∗) = {∗}, for every c ∈ Θn (with n > 0) and c1 = {∗}, for every c ∈ Θ0 . Proof. Let A be a multialgebra over Θ, then !A : A → {∗}, such that for every a ∈ A, !A (a) = ∗ is the only (full) homomorphism from A to 1. Proposition 2.5.8. The category MAlg(Θ) has arbitrary products. Proof. Let {Ai }i∈I be a family of multialgebras over Θ. If I = ∅, since the multialgebra 1 = h{∗}, σ1 i is the terminal object in MAlg(Θ), then the result is obvious. Now, assume that I 6= ∅. Consider the product A = i∈I Ai as in Definition 2.5.6. Then, πi [cA (~a)] = Q πi [ i∈I (cAi (πi (a1 ), . . . , πi (an )))] = cAi (πi (a1 ), . . . , πi (an )). In particular, we have πi [cA ] = Q πi [ i∈I cAi ] = cAi . Therefore, πi is a (full) homomorphism from A to Ai . Q

Let B be an object in MAlg(Θ), let µi : B → Ai be a morphism in MAlg(Θ), for every i ∈ I and let δ : B → A such that δ : |B| → |A| be the function defined by Q δ(b) = i∈I µi (b) for every b ∈ |B|, then: (i) If b ∈ |B|, πi ◦ δ(b) = πi (δ(b)) = πi ( i∈I µi (b)) = µi (b), so πi ◦ δ = µi . Q Q Q (ii) δ[cB (~b)] = i∈I µi (cB (~b)) ⊆ i∈I cAi (µi (b1 ), . . . , µi (bn )) = i∈I cAi (πi ◦ δ(b1 ), . . . , πi ◦ Q δ(bn )) = i∈I cAi (πi (δ(b1 )), . . . , πi (δ(bn ))) = cA (δ(b1 ), . . . , δ(bn )). In particular, if c ∈ Θ0 , Q Q δ[cB ] = i∈I µi [cB ] ⊆ i∈I cAi = cA , so δ is a homomorphism in MAlg(Θ). (iii) Supose that δ 0 : B → A is a morphism in MAlg(Θ) such that πi ◦ δ 0 = µi . If b ∈ |B|, Q Q Q Q δ(b) = i∈I µi (b) = i∈I (πi ◦ δ 0 (b)) = i∈I πi (δ 0 (b)) = i∈I (δ 0 (b))(i) = δ 0 (b), so δ is unique. By (i), (ii) and (iii), we have that hA, {πi }i∈I i is the product in MAlg(Θ) of the family {Ai }i∈I . Q

Moreover, the following useful result holds in MAlg(Θ): Proposition 2.5.9 (Epi-mono factorization). Let A = hA, σA i and B = hB, σB i be two multialgebras over Θ, and let f : A → B be a homomorphism in MAlg(Θ). Let f¯ : A → f [A] be the mapping given by f¯(x) = f (x) for every x ∈ A, and let g : f [A] → B be the inclusion map. Then f¯ and g are homomorphisms f¯ : A → f (A) and g : f (A) → B such that f¯ is an epimorphism in MAlg(Θ), g is a monomorphism in MAlg(Θ), and


64

f = g ◦ f¯. A

f

f¯

'

/

BO ?

g

f (A)

Moreover, if f is (one-to-one) injective (as a function) then f¯ is an isomorphism in MAlg(Θ). Proof. (i) If ~a ∈ An (n > 0), f¯[cA (~a)] = f [cA (~a)]. By Proposition 2.2.13, f [cA (~a)] ⊆ S {f [cA (~a)] : ~a ∈ f −1 (f (a1 ), . . . , f (an ))} = cf (A) (f (a1 ), . . . , f (an )) = cf (A) (f¯(a1 ), . . . , f¯(an )). In particular, f¯[cA ] = f [cA ] = cf (A) . Therefore, f¯ is a homomorphism from A to f (A). (ii) By Proposition 2.2.13 and by Proposition 2.5.1, we have that g is a homomorphism from f (A) to B. (iii) Note that f¯ is onto: If b ∈ f [A] = {f (a) : a ∈ A} = {f¯(a) : a ∈ A}, there is a ∈ A such that f¯(a) = b and by Proposition 2.5.5 we have that f¯ is an epimorphism in MAlg(Θ). (iv) Note that g is one-to-one: If b1 , b2 ∈ f [A] and g(b1 ) = g(b2 ), then b1 = g(b1 ) = g(b2 ) = b2 . So, by Proposition 2.5.4 we have that g is a monomorphism in MAlg(Θ). (v) If a ∈ A, g ◦ f¯(a) = g(f¯(a)) = f¯(a) = f (a), so f = g ◦ f¯. (vi) Note that f¯ is one-to-one: if a1 , a2 ∈ A, f (a1 ) = f¯(a1 ) = f¯(a2 ) = f (a2 ). Since by hypothesis f is one-to-one, then a1 = a2 . By (iii) we have also that f¯ is onto. By (i), If ~a ∈ An (n > 0), f¯[cA (~a)] ⊆ cf (A) (f¯(a1 ), . . . , f¯(an )), on other hand cf (A) (f¯(a1 ), . . . , f¯(an )) = S cf (A) (f (a1 ), . . . , f (an )) = {f [cA (~a)] : ~a ∈ f −1 (f (a1 ), . . . , f (an ))} = f [cA (~a)] = f¯[cA (~a)]. Therefore, f¯ is a full homomorphism and by Proposition 2.5.3 f¯ is an isomorphism in MAlg(Θ).

In the next chapters, we will show the use of the multialgebras in the algebraization of logical systems.

65

3 Non-deterministic semantics normal modal logics

for

non-

In this chapter we will introduce swap structures for some non-normal modal systems and we will use the Lindenbaum-Tarski swap structures to obtain completeness theorems w.r.t Hilbert-style version of these systems. To begin with, however, we will introduce, formally, these modal logics.

3.1 The Systems Tm, T4m, T45m, TBm and Dm John T. Kearns in the paper “Modal semantics without possible worlds” published in 1981 (KEARNS, 1981) proposes a semantics of four values for modal logics without using the concept of possible worlds. That is, different of the well-known Kripke relational semantics. In this semantics, non-deterministic matrices are used to characterize the logical operators. It is interesting to observe that this is one of the first relevant applications of non-deterministic matrices in the literature. The truth-values consider by Kearns are: T : Necessary truth t: Contingent truth f : Contingent falsity F : Necessary falsity In 1988, in the paper called “A semantics for modal calculi” (IVLEV, 1988), J. V. Ivlev presented a semantics of non-deterministic matrices of four values, in order to semantically characterize a family of weaker modal logics in which the necessitation rule1 is not valid. Ivlev, to define the non-deterministic matrices of four values, considers the following truth-values: tn : Necessary truth 1

For more information about modal logic and about the necessitation rule, see for instance (CHELLAS, 1980).

Chapter 3. Non-deterministic semantics for non-normal modal logics

66

tc : Possible truth f c : Possible falsity f i : Necessary falsity The set of designated truth-values is {tn , tc } and the set of non-designated truth-values is {f i , f c }. If we consider the signature Σ0 = {¬, →, } and tn = T , tc = t, f c = f and f i = F , then the Ivlev2 and Kearns3 (four-valued) non-deterministic matrix semantics coincide. They are given by: →

tn

tc

fc

fi

α

¬α

α

α

tn

tn

tc

fc

fi

tn

fi

tn

tn /tc

tc

tn

tn /tc

fc

fc

tc

fc

tc

f c /f i

fc

tn

tn /tc

tn /tc

tc

fc

tc

fc

f c /f i

fi

tn

tn

tn

tn

fi

tn

fi

f c /f i

The operators ∨ and ♦ can be defined, as in the classical case, respectively by α ∨ β = ¬α → β and ♦α = ¬¬α, so they produce the following matrices: ∨

tn

tc

fc

fi

α

♦α

tn

tn

tn

tn

tn

tn

tn /tc

tc

tn

tn /tc

tn /tc

tc

tc

tn /tc

fc

tn

tn /tc

fc

fc

fc

tn /tc

fi

tn

tc

fc

fi

fi

f c /f i

A Hilbert-style deductive system for Sa + and of T was introduced in (CONIGLIO; CERRO; PERON, 2015) with the notation Tm. A slightly different version will be introduced here, following the modifications suggested in (OMORI; SKURT, 2016; CONIGLIO; CERRO; PERON, 2016). Definition 3.1.1. The system Tm over Σ0 = {¬, →, } is composed of the following axiom schemes: α → (β → α) (Ax1) (α → (β → σ)) → ((α → β) → (α → σ)) (Ax2) 2 3

Semantics matrices for the system Sa +, see (IVLEV, 1988). Semantics matrices for the system T , see (KEARNS, 1981).


67

(¬β → ¬α) → ((¬β → α) → β) (Ax3) (α → β) → (α → β) (K) (α → β) → (¬β → ¬α) (K1) ¬¬(α → β) → (α → ¬¬β) (K2) ¬α → (α → β) (M 1) β → (α → β) (M 2) ¬(α → β) → ¬β (M 3) ¬(α → β) → α (M 4) α → α (T ) α → ¬¬α (DN 1) ¬¬α → α (DN 2) And by the rule of inference: α, α → β (M P ) β The systems T4m, T45m, TBm and Dm were introduced in (CONIGLIO; CERRO; PERON, 2015). Now, to define them, we consider the following axioms: α → α (4) ¬¬α → α (5) ¬¬α → α (B) α → ¬¬α (D) The versions of Hilbert-style deductive systems for T4m, T45m and TBm are given by: T4m = Tm ∪ {(4)} T45m = T4m ∪ {(5)} TBm = Tm ∪ {(B)} The non-deterministic matrix semantics for the systems T4m, T45m and TBm are given by the same multioperation → and by the same operation ¬ of Tm only by changing the multioperation , respectively to:


68

α

T4m α

α

T45m α

α

TBm α

tn

tn

tn

tn

tn

tn /tc

tc

f c /f i

tc

fi

tc

f c /f i

fc

f c /f i

fc

fi

fc

fi

fi

f c /f i

fi

fi

fi

fi

The operation T45m is the same operation proposed by Kearns in (KEARNS, 1981) for the system S5 and by Ivlev for the system Sb + in (IVLEV, 1988). And, by taking (f i ) = {f i } instead of (f i ) = {f c , f i }, then the multioperation proposed by Kearns (KEARNS, 1981) in order to interpret the system S4 and the multioperation T4m of T4m will be the same too. Deontic logics are known to interpret, intuitively, the operators and ♦, respectively, by it is obligatory that and by it is permitted that. In (CONIGLIO; CERRO; PERON, 2015) it was proposed that, under this interpretation, to accept a weaker principle α → ♦α instead of principles like α → α and α → ♦α would be more convenient. And so a new (weakly) deontic system characterized by six-valued non-deterministic matrix semantics was obtained. This new system was called Dm. In Dm a proposition p is said to be: • fulfilled whenever either it is obligatory and it is the case, or it is forbidden and it is not the case (that is, it is the case of its negation). In symbols: (p ∧ p) ∨ (¬p ∧ ¬p). • infringed whenever either it is obligatory and it is not the case, or it is forbidden and it is the case. In symbols: (p ∧ ¬p) ∨ (¬p ∧ p). • optional if it is neither obligatory nor forbidden. In symbols: ¬p ∧ ¬¬p. It generates six truth-values as follows: T + : p is fulfilled; C + : p is optional; F + : p is infringed; T − : ¬p is infringed; C − : ¬p is optional and F − : ¬p is fulfilled.


69

The versions of Hilbert-style deductive system for Dm is given by: Dm = Tm ∪ {(D)} − {(T )}

→

Dm

And the semantics matrices for the system Dm is given by the multioperations and Dm and by the operation ¬Dm : →Dm

T+

T−

C+

C−

F+

F−

T+

T+

T−

C+

C−

F+

F−

T−

T+

T+

C+

C+

F+

F+

C+

T+

T−

{T + , C + }

{T − , C − }

C+

C−

C−

T+

T+

{T + , C + }

{T + , C + }

C+

C+

F+

T+

T−

T+

T−

T+

T−

F−

T+

T+

T+

T+

T+

T+

α

¬Dm α

α

Dm α

T+

F−

T+

{T + , C + , F + }

T−

F+

T−

{T + , C + , F + }

C+

C−

C+

{T − , C − , F − }

C−

C+

C−

{T − , C − , F − }

F+

T−

F+

{T − , C − , F − }

F−

T+

F−

{T − , C − , F − }

Let • {T + , T − } is the set interpreted as obligatory; • {C + , C − } is the set interpreted as optional; • {F + , F − } is the set interpreted as forbidden; • + = {T + , C + , F + } is the set of designated truth-values and • − = {T − , C − , F − } is the set of non-designated truth-values. In the next section, we will introduce swap structures for the systems Tm, T4m, T45m, TBm and Dm.


70

3.2 Swap structures for Tm, T4m, T45m, TBm and Dm In this chapter we will introduce an original class of swap structures as semantics for a family of non-normal modal systems. Through swap structures we are proposing a class of multialgebras, whose elements are triples in a particular Boolean algebra and the operations change of place (swap) some components of the triples. The elements of a given swap structure are called snapshots4 . In the swap structures, we ‘swap’ in a certain manner the components of the snapshots and this justifies the name adopted for these multialgebras. The consequence relation over swap structures will be presented by means of logical matrices, but the swap structures for the modal systems presented here are more general than the corresponding non-deterministic matrices. Intuitively, in this Chapter, the triples for any formula α are (α, α, ¬α). After we define swap structures for these modal logics, we will prove soundness and completeness theorems of the Hilbert calculi defining these non-normal modal systems with respect to such non-deterministic matrix semantics. In order to obtain the completeness theorem, we will to apply the method of Lindenbaum-Tarski swap structures, that generalizes in a quite natural way the classical Lindenbaum-Tarski method, allowing to deal with logics which are not algebraizable in the usual sense as is the case of the modal logics presented in the previous sections of this chapter. Definition 3.2.1 (Swap structures for Tm). Let A = hA, ∨, ∧, →, 0, 1i be a Boolean algebra and let BTm = {(a1 , a2 , a3 ) ∈ A3 : a2 ≤ a1 and a1 ∧ a3 = 0}. A A swap structure for Tm over A is any multialgebra B = hB, →, ¬, i over Σ0 = {→, ¬, } such that B ⊆ BTm and the multioperations satisfy the following, for A every (a1 , a2 , a3 ) and (b1 , b2 , b3 ) in B: (i) (a1 , a2 , a3 ) → (b1 , b2 , b3 ) = {(c1 , c2 , c3 ) ∈ B : c1 = a1 → b1 , c3 = a2 ∧b3 and a3 ∨b2 ≤ c2 ≤ (a1 → b1 ) ∧ (a2 → b2 ) ∧ (b3 → a3 )} (ii) ¬(a1 , a2 , a3 ) = {(¬a1 , a3 , a2 )} (iii) (a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ B : c1 = a2 } 4

This terminology is inspired by its use in computer systems to refer to states.


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Tm The unique swap structure for Tm, with domain BTm A , will be denoted by BA .

The next proposition shows that the implication in Tm is a well-defined multioperation: (a1 , a2 , a3 ) → (b1 , b2 , b3 ) 6= ∅ for (a1 , a2 , a3 ), (b1 , b2 , b3 ) ∈ BTm A . Proposition 3.2.2. : If (a1 , a2 , a3 ), (b1 , b2 , b3 ) ∈ BTm A , then a3 ∨ b2 ≤ (a1 → b1 ) ∧ (a2 → b2 ) ∧ (b3 → a3 ). Proof. Let be (a1 , a2 , a3 ), (b1 , b2 , b3 ) ∈ BTm A , As a1 ∧ a3 = 0, then a3 ≤ ¬a1 . But, as a2 ≤ a1 , then ¬a1 ≤ ¬a2 and, therefore, a3 ≤ ¬a2 . So, a3 ∨ b2 ≤ ¬a2 ∨ b2 = a2 → b2 (3.1) It also holds that (a3 ∨ b2 ) ∧ a1 = (a3 ∧ a1 ) ∨ (b2 ∧ a1 ) = 0 ∨ (b2 ∧ a1 ) = b2 ∧ a1 ≤ b2 ≤ b1 ≤ a1 → b1 . So, a3 ∨ b 2 ≤ a1 → b 1 (3.2) On the other hand, (a3 ∨ b2 ) ∧ b3 = (a3 ∧ b3 ) ∨ (b2 ∧ b3 ) ≤ (a3 ∧ b3 ) ∨ (b1 ∧ b3 ) = (a3 ∧ b3 ) ∨ 0 = a3 ∧ b3 ≤ a3 . So, a3 ∨ b2 ≤ b3 → a3 (3.3) From (3.1), (3.2) and (3.3), we have a3 ∨ b2 ≤ (a1 → b1 ) ∧ (a2 → b2 ) ∧ (b3 → a3 ). Proposition 3.2.3. : If (a1 , a2 , a3 ), (b1 , b2 , b3 ) ∈ BTm A , c1 = a1 → b1 and c3 = a2 ∧ b3 , then c3 ∧ c1 = 0. Proof. (a2 ∧ b3 ) ∧ (a1 → b1 ) = (a2 ∧ b3 ) ∧ (¬a1 ∨ b1 ) = (b3 ∧ (a2 ∧ ¬a1 )) ∨ ((b3 ∧ b1 ) ∧ a2 ). Since, a2 ∧ ¬a1 = (a2 ∧ a1 ) ∧ ¬a1 = a2 ∧ 0 = 0, then (b3 ∧ (a2 ∧ ¬a1 )) ∨ ((b3 ∧ b1 ) ∧ a2 ) = (b3 ∧ 0) ∨ (0 ∧ a2 ) = 0. In the particular case of the two-element Boolean algebra A2 with domain A2 = {0, 1}, eight triples are possible: (1, 1, 1), (1, 0, 1), (0, 1, 0), (0, 1, 1), (1, 1, 0), (1, 0, 0), (0, 0, 0) and (0, 0, 1). However, a triple (a1 , a2 , a3 ) in A32 belongs to BTm A2 , whenever satisfies the conditions a2 ≤ a1 and a1 ∧ a3 = 0. By the first condition, (0, 1, 0) and (0, 1, 1) are discarded and by the second condition, the triples (1, 1, 1) and (1, 0, 1) are discarded too. So, BTm A2 = {(1, 1, 0), (1, 0, 0), (0, 0, 0), (0, 0, 1)} and if we denote the values (1, 1, 0), (1, 0, 0), (0, 0, 0) and (0, 0, 1), respectively by tn , tc , f c and f i then the truth tables of Ivlev for Sa + are a particular case of our definition of swap structure for Tm. Definition 3.2.4 (Swap structures for T4m). A swap structure for T4m over A is a swap structure for Tm such that the multioperation is given by (a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ B : c1 = a2 and a2 ≤ c2 } for every (a1 , a2 , a3 ) in B.


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Definition 3.2.5 (Swap structures for T45m). A swap structure for T45m over A is a swap structure for Tm such that the multioperation is given by (a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ B : c1 = a2 , a2 ≤ c2 and c3 ∨ c1 = 1} for every (a1 , a2 , a3 ) in B. Definition 3.2.6 (Swap structures for TBm). A swap structure for TBm over A is a swap structure for Tm such that the multioperation is given by (a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ B : c1 = a2 and a1 ∨ c3 = 1} for every (a1 , a2 , a3 ) in B. Definition 3.2.7 (Swap structures for Dm). Let A = hA, ∨, ∧, →, 0, 1i be a Boolean algebra and let BDm = {(a1 , a2 , a3 ) ∈ A3 : a2 ∧ a3 = 0}. A A swap structure for Dm over A is any multialgebra B = hB, →, ¬, i over Σ0 = {→, ¬, } such that B ⊆ BDm and the multioperations satisfy the following, A for every (a1 , a2 , a3 ) and (b1 , b2 , b3 ) in B: (i) (a1 , a2 , a3 ) → (b1 , b2 , b3 ) = {(c1 , c2 , c3 ) ∈ B : c1 = a1 → b1 , c3 = a2 ∧b3 and a3 ∨b2 ≤ c2 ≤ (a2 → b2 ) ∧ (b3 → a3 )} (ii) ¬(a1 , a2 , a3 ) = {(¬a1 , a3 , a2 )} (iii) (a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ B : c1 = a2 } The unique swap structure for Dm, with domain BDm A , will be denoted by Dm BA .

As in the case of Tm, it will be shown that the implication (a1 , a2 , a3 ) → (b1 , b2 , b3 ) in Dm returns a non-empty set, for (a1 , a2 , a3 ), (b1 , b2 , b3 ) ∈ BDm A : Proposition 3.2.8. : If (a1 , a2 , a3 ), (b1 , b2 , b3 ) ∈ BDm A , then a3 ∨ b2 ≤ (a2 → b2 ) ∧ (b3 → a3 ). Proof. Let be (a1 , a2 , a3 ), (b1 , b2 , b3 ) ∈ BDm A . We have a3 ∧ ¬a2 = a3 ∧ ¬a2 a3 ∧ ¬a2 = 0 ∨ (a3 ∧ ¬a2 )


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a3 ∧ ¬a2 = (a3 ∧ a2 ) ∨ (a3 ∧ ¬a2 ) a3 ∧ ¬a2 = a3 ∧ (a2 ∨ ¬a2 ) a3 ∧ ¬a2 = a3 ∧ 1 a3 ∧ ¬a2 = a3 . But, a3 ∧ ¬a2 = a3 iff a3 ≤ ¬a2 . Then a3 ∨ b2 ≤ ¬a2 ∨ b2 = a2 → b2

(3.4)

On the other hand, b2 ∧ ¬b3 = b2 ∧ ¬b3 b2 ∧ ¬b3 = 0 ∨ (b2 ∧ ¬b3 ) b2 ∧ ¬b3 = (b2 ∧ b3 ) ∨ (b2 ∧ ¬b3 ) b2 ∧ ¬b3 = b2 ∧ (b3 ∨ ¬b3 ) b2 ∧ ¬b3 = b2 ∧ 1 b2 ∧ ¬b3 = b2 . But, b2 ∧ ¬b3 = b2 iff b2 ≤ ¬b3 . Then a3 ∨ b2 ≤ a3 ∨ ¬b3 = b3 → a3

(3.5)

From (3.4) and (3.5), we have a3 ∨ b2 ≤ (a2 → b2 ) ∧ (b3 → a3 ). Proposition 3.2.9. : If (a1 , a2 , a3 ), (b1 , b2 , b3 ) ∈ BDm and c3 = a2 ∧ b3 , then there is c2 A such that c2 ∧ c3 = 0. Proof. Let be c2 = a3 ∨ b2 , then (a3 ∨ b2 ) ∧ (a2 ∧ b3 ) = (a2 ∧ b3 ∧ a3 ) ∨ (a2 ∧ b3 ∧ b2 ) = (0 ∧ b3 ) ∨ (a2 ∧ 0) = 0 ∨ 0 = 0. Notation 3.2.10. If L ∈ {Tm, T4m, T45m, TBm, Dm}, then we denote by KL the class of swap structures for the system L . Definition 3.2.11. Let L ∈ {Tm, T4m, T45m, TBm, Dm} and DB = {(z1 , z2 , z3 ) ∈ |B| : z1 = 1} for each B ∈ KL . The non-deterministic matrix associated to B is M(B) = hB, DB i. If A = A2 , then M(BA2 ) = hBA2 , DBA2 i.


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Notation 3.2.12. If L ∈ {Tm, T4m, T45m, TBm, Dm}, then we denote by M at(KL ) = {M(B) : B ∈ KL } the class of non-deterministic matrix associated to swap structure to the system L. Recall the semantics associated to non-deterministic matrices where the relation ∆ |=M α is the consequence relation defined over a non-deterministic matrix M (see Definitions 2.1.4, 2.1.5): Definition 3.2.13. For L ∈ {Tm, T4m, T45m, TBm, Dm}, let ∆ ∪ {α} ⊆ F or(Σ0 ) be a set of formulas of L. We say that α is a consequence of ∆ in the class M at(KL ) of non-deterministic matrices, and we denote it by ∆ |=M at(KL ) α, if ∆ |=M α for every M ∈ M at(KL ). In particular, α is valid in M at(KL ), denoted by |=M at(KL ) α, if it is valid in every M ∈ M at(KL ). Theorem 3.2.14. (Deduction theorem) For L ∈ {Tm, T4m, T45m, TBm, Dm}, let ∆ ∪ {α, β} be a set of formulas in L. Then ∆ ∪ {α} `L β iff ∆ `L α → β. Proof. Since in the systems Tm, T4m, T45m, TBm and Dm the necessitation rule is not valid, then this classical result follows from the fact that these logics are axiomatic extensions of classical logic. A proof can be found in (MENDELSON, 1987). Lemma 3.2.15. Let πi : A3 → A be the function projection on the ith coordinate of A3 , such that πi ((a1 , a2 , a3 )) = ai for i = 1, 2, 3. Let α, β ∈ F or(Σ0 ) and let v be any valuation for a swap structure for L ∈ {Tm, T4m, T45m, TBm, Dm}. Then, i) π1 (v(α → β)) = π1 (v(α)) → π1 (v(β)); ii) π1 (v(¬α)) = ¬π1 (v(α)). Proof. i) v(α → β) ∈ v(α) → v(β) = {(c1 , c2 , c3 ) ∈ B : c1 = π1 (v(α)) → π1 (v(β)), c3 = π2 (v(α)) ∧ π3 (v(β)) and π3 (v(α)) ∨ π2 (v(β)) ≤ c2 ≤ (π1 (v(α)) → π1 (v(β))) ∧ (π2 (v(α)) → π2 (v(β))) ∧ (π3 (v(β)) → π3 (v(α)))}. So, π1 (v(α → β)) = π1 (v(α)) → π1 (v(β)). ii) v(¬α) ∈ ¬v(α) = {(¬π1 (v(α)), π3 (v(α)), π2 (v(α)))}. So, π1 (v(¬α)) = ¬π1 (v(α)).

Lemma 3.2.16. Let πi : A3 → A be the function projection on the ith coordinate of A3 , such that πi ((a1 , a2 , a3 )) = ai for i = 1, 2, 3. Let α, β ∈ F or(Σ0 ) and let v be any valuation for a swap structure for L ∈ {Tm, T4m, T45m, TBm, Dm}. So, the following conditions are equivalent:


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i) v(α → β) ∈ DB ; ii) π1 (v(α)) ≤ π1 (v(β)); iii) π1 (v(α)) → π1 (v(β)) = 1. Proof. (i) ⇔ (iii) : v(α → β) ∈ DB iff π1 (v(α → β)) = 1. But, by Lemma 3.2.15, we have π1 (v(α → β)) = π1 (v(α)) → π1 (v(β)) and, therefore π1 (v(α)) → π1 (v(β)) = 1. On the other hand, it is well-known that a ≤ b iff a → b = 1 in any Boolean algebra. This completes the proof. Theorem 3.2.17. (Soundness) If L ∈ {Tm, T4m, T45m, TBm, Dm} and α ∈ F or(Σ0 ), then `L α ⇒ M at(KL ) α. Proof. Let v be a valuation for a swap structure B for L. We will prove by induction over the length of the deduction of α in L, that v(α) ∈ DB . If there is only one formula in the deduction of α, this formula is α itself. Then α can only be an axiom of L. So, we must verify that the result is true for every axiom of L: To simplify the proof, we will use Lemma 3.2.16 and the following notation: If α is a formula and v is a valuation, we’ll write |α|1 , |α|2 and |α|3 instead of, respectively, π1 (v(α)), π2 (v(α)) and π3 (v(α)). The task of checking the axioms will be divided into three parts: Part 1: If α is an axiom in {(Ax1), (Ax2), (Ax3), (K2), (M 1), (M 2), (M 3), (M 4), (DN 1), (DN 2)}, then the proof is the same for any system L. See below: • If α is of the form δ → (β → δ): Note that |β → δ|1 = |β|1 → |δ|1 = ¬|β|1 ∨ |δ|1 , so |δ|1 ≤ ¬|β|1 ∨ |δ|1 = |β → δ|1 . • If α is of the form (δ → (β → σ)) → ((δ → β) → (δ → σ)): Note that |(δ → β) → (δ → σ)|1 = |δ → β|1 → |δ → σ|1 = (|δ|1 → |β|1 ) → (|δ|1 → |σ|1 ) = ¬(|δ|1 → |β|1 ) ∨ (|δ|1 → |σ|1 ) = ¬(¬|δ|1 ∨ |β|1 ) ∨ (¬|δ|1 ∨ |σ|1 ) = (|δ|1 ∧ ¬|β|1 ) ∨ (¬|δ|1 ∨ |σ|1 ) = (|δ|1 ∨ (¬|δ|1 ∨ |σ|1 )) ∧ (¬|β|1 ∨ (¬|δ|1 ∨ |σ|1 )) = (1 ∨ |σ|1 ) ∧ (¬|δ|1 ∨ (¬|β|1 ∨ |σ|1 )) = 1 ∧ (|δ|1 → (|β|1 → |σ|1 )) = |δ|1 → (|β|1 → |σ|1 ) = |δ → (β → σ)|1 . In particular, |δ → (β → σ)|1 ≤ |(δ → β) → (δ → σ)|1 . • If α is of the form (¬β → ¬δ) → ((¬β → δ) → β): Note that |(¬β → δ) → β)|1 = (¬|β|1 → |δ|1 ) → |β|1 = ¬(¬|β|1 → |δ|1 ) ∨ |β|1 = ¬(|β|1 ∨ |δ|1 ) ∨ |β|1 = (¬|β|1 ∧ ¬|δ|1 ) ∨ |β|1 = (¬|β|1 ∨ |β|1 ) ∧ (¬|δ|1 ∨ |β|1 ) =


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1 ∧ (¬|δ|1 ∨ |β|1 ) = |δ|1 → |β|1 = ¬|β|1 → ¬|δ|1 = |¬β → ¬δ|1 . In particular, |¬β → ¬δ|1 ≤ |(¬β → δ) → β)|1 . • If α is of the form ¬¬(δ → β) → (δ → ¬¬β): Note that |¬¬(δ → β)|1 = ¬|¬(δ → β)|1 = ¬|¬(δ → β)|2 = ¬|(δ → β)|3 = ¬(|δ|2 ∧ |β|3 ) = ¬|δ|2 ∨ ¬|β|3 = |δ|2 → ¬|β|3 . And |δ → ¬¬β|1 = |δ|1 → |¬¬β|1 = |δ|2 → ¬|¬β|1 = |δ|2 → ¬|¬β|2 = |δ|2 → ¬|β|3 . In particular, |¬¬(δ → β)|1 ≤ |δ → ¬¬β|1 . • If α is of the form ¬δ → (δ → β): Note that |¬δ|1 = ¬|δ|2 = |δ|3 and |(δ → β)|1 = |δ → β|2 . But, |δ|3 ≤ |δ|3 ∨|β|2 ≤ |δ → β|2 . So, |¬δ|1 ≤ |(δ → β)|1 . • If α is of the form β → (δ → β): Note that |β|1 = |β|2 and |(δ → β)|1 = |δ → β|2 . But, |β|2 ≤ |δ|3 ∨|β|2 ≤ |δ → β|2 . So, |β|1 ≤ |(δ → β)|1 . • If α is of the form ¬(δ → β) → ¬β: Note that |¬β|1 = |¬β|2 = |β|3 and |¬(δ → β)|1 = |¬(δ → β)|2 = |δ → β|3 = |δ|2 ∧ |β|3 . So, |¬(δ → β)|1 = |δ|2 ∧ |β|3 ≤ |β|3 = |¬β|1 . • If α is of the form ¬(δ → β) → δ: Note that |δ|1 = |δ|2 and |¬(δ → β)|1 = |¬(δ → β)|2 = |δ → β|3 = |δ|2 ∧ |β|3 . So, |¬(δ → β)|1 = |δ|2 ∧ |β|3 ≤ |δ|2 = |δ|1 . • If α is of the form δ → ¬¬δ or α is of the form ¬¬δ → δ: Note that |δ|1 = |δ|2 and |¬¬δ|1 = |¬¬δ|2 = |¬δ|3 = |δ|2 . In particular, |δ|1 ≤ |¬¬δ|1 and |¬¬δ|1 ≤ |δ|1 . Part 2: If α is an axiom in {(K), (K1)}, then the proof is the same for any system L ∈ {Tm, T4m, T45m, TBm}. So, we will be divided the proof of each one into two parts. • If α is of the form (δ → β) → (δ → β): If L ∈ {Tm, T4m, T45m, TBm}: From |(δ → β)|1 = |δ → β|2 and we have |δ|3 ∨ |β|2 ≤ |δ → β|2 ≤ (|δ|1 → |β|1 ) ∧ (|δ|2 → |β|2 ) ∧ (|β|3 → |δ|3 ). On the other hand, |δ → β|1 = |δ|1 → |β|1 = |δ|2 → |β|2 . So, |(δ → β)|1 ≤ (|δ|1 → |β|1 ) ∧ |δ → β|1 ∧ (|β|3 → |δ|3 ). Therefore, |(δ → β)|1 ≤ |δ → β|1 .


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If L = Dm: From |(δ → β)|1 = |δ → β|2 and we have |δ|3 ∨ |β|2 ≤ |δ → β|2 ≤ (|δ|2 → |β|2 ) ∧ (|β|3 → |δ|3 ). On the other hand, |δ → β|1 = |δ|1 → |β|1 = |δ|2 → |β|2 . So, |(δ → β)|1 ≤ |δ → β|1 ∧ (|β|3 → |δ|3 ). Therefore, |(δ → β)|1 ≤ |δ → β|1 . • If α is of the form (δ → β) → (¬β → ¬δ): If L ∈ {Tm, T4m, T45m, TBm}: From |(δ → β)|1 = |δ → β|2 and we have |δ|3 ∨ |β|2 ≤ |δ → β|2 ≤ (|δ|1 → |β|1 ) ∧ (|δ|2 → |β|2 ) ∧ (|β|3 → |δ|3 ). On the other hand, |¬β → ¬δ|1 = |¬β|1 → |¬δ|1 = |¬β|2 → |¬δ|2 = |β|3 → |δ|3 . So, |(δ → β)|1 ≤ (|δ|1 → |β|1 ) ∧ (|δ|2 → |β|2 ) ∧ |¬β → ¬δ|1 . Therefore, |(δ → β)|1 ≤ |¬β → ¬δ|1 . If L = Dm: From |(δ → β)|1 = |δ → β|2 and we have |δ|3 ∨ |β|2 ≤ |δ → β|2 ≤ (|δ|2 → |β|2 ) ∧ (|β|3 → |δ|3 ). On the other hand, |¬β → ¬δ|1 = |¬β|1 → |¬δ|1 = |¬β|2 → |¬δ|2 = |β|3 → |δ|3 . So, |(δ → β)|1 ≤ (|δ|2 → |β|2 ) ∧ |¬β → ¬δ|1 . Therefore, |(δ → β)|1 ≤ |¬β → ¬δ|1 . Part 3: In this part, we will check the specific axioms of each system. See below: • If α is of the form δ → δ and L ∈ {Tm, T4m, T45m, TBm}: As |δ|1 = |δ|2 then, by definition of BTm A , we have |δ|2 ≤ |δ|1 . So, |δ|1 ≤ |δ|1 . • If α is of the form δ → ¬¬δ and L = Dm: Note that |δ|1 = |δ|2 and |¬¬δ|1 = ¬|¬δ|1 = ¬|¬δ|2 = ¬|δ|3 and by definition of BDm A , we have |δ|2 ∧ |δ|3 = 0. From this |δ|2 ≤ ¬|δ|3 . Therefore, |δ|1 ≤ |¬¬δ|1 . • If α is of the form ¬¬δ → δ and L = TBm: Note that |¬¬δ|1 = ¬|¬δ|1 = ¬|¬δ|2 = ¬|δ|3 , such that |δ|1 ∨ |δ|3 = 1 (by Definition 3.2.6). But, then ¬|δ|3 → |δ|1 = 1. And so, ¬|δ|3 ≤ |δ|1 . Therefore, |¬¬δ|1 ≤ |δ|1 . • If α is of the form ¬¬δ → δ and L = T45m: Note that |¬¬δ|1 = ¬|¬δ|1 = ¬|¬δ|2 = ¬|δ|3 , such that |δ|1 ∨ |δ|3 = 1 (by Definition 3.2.5). But, then ¬|δ|3 → |δ|1 = 1. And so, ¬|δ|3 ≤ |δ|1 . Therefore, |¬¬δ|1 ≤ |δ|1 .


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• If α is of the form δ → δ and L ∈ {T4m, T45m}: As |δ|1 = |δ|2 such that (by Definitions 3.2.4 and 3.2.5) |δ|2 ≤ |δ|2 and |δ|1 = |δ|2 . So, |δ|1 ≤ |δ|1 . Suppose the result is true for every formula that has a shorter deduction length than the deduction of α. • If α is an axiom, we have already proved that v(α) ∈ DB , for all valuation v for L ∈ {Tm, T4m, T45m, TBm, Dm}. • If α follows from two previous formulas by (M P ), so these formulas must be the form β and β → α. By hypothesis, we have `L β and `L β → α. Then, by hypothesis of induction, we have M at(KL ) β and M at(KL ) β → α. Let v be a valuation. By definition, we have M at(KL ) β iff v(β) ∈ DB iff π1 (v(β)) = 1 and M at(KL ) β → α iff v(β → α) ∈ DB iff π1 (v(β → α)) = 1 and by Lemma 3.2.15 we have π1 (v(β → α)) = π1 (v(β)) → π1 (v(α)). So, π1 (v(β)) → π1 (v(α)) = 1 and as π1 (v(β)) = 1, then π1 (v(α)) = 1. Therefore, v(α) ∈ DB for every valuation and so M at(KL ) α.

Theorem 3.2.18. (Strong soundness) If L ∈ {Tm, T4m, T45m, TBm, Dm} and ∆ ∪ {α} ⊆ F or(Σ0 ), then ∆ `L α ⇒ ∆ M at(KL ) α. Proof. If ∆ = ∅, the result has already been proved in the Theorem 3.2.17. If ∆ 6= ∅. Let v be a valuation such that v(δ) ∈ DB for every δ ∈ ∆. Let β1 , . . . , βn be all the elements of ∆ that appear in a deduction of α from ∆. As hypothesis so, we have β1 , . . . , βn `L α and, by successive applications of the Deduction Theorem, we have `L β1 → (. . . → (βn → α) . . .). However, by the Theorem 3.2.17, we have M at(KL ) β1 → (. . . → (βn → α) . . .). But, M at(KL ) β1 → (. . . → (βn → α) . . .) ⇔ (β1 → (. . . → (βn → α) . . .)) ∈ DB ⇒ π1 (v(β1 → (. . . → (βn → α) . . .))) = 1. On the other hand, by Lemma 3.2.15, π1 (v(β1 → (. . . → (βn → α) . . .))) = π1 (v(β1 )) → (. . . → ((π1 (v(βn )) → π1 (v(α))) . . .). So, π1 (v(β1 )) → (. . . → ((π1 (v(βn )) → π1 (v(α))) . . .) = 1. But, by hypothesis v(βi ) ∈ DB , for all βi with (1 ≤ i ≤ n) and consequently π1 (v(βi )) = 1 for all βi with (1 ≤ i ≤ n). So, π1 (v(α)) = 1, but π1 (v(α)) = 1 ⇔ v(α) ∈ DB . Then, ∆ M at(KL ) α.

Definition 3.2.19. If L ∈ {Tm, T4m, T45m, TBm, Dm} and ∆ is a theory in L. Then, ≡∆ is a relation between formulas of L, such that: α ≡∆ β iff ∆ `L α → β and ∆ `L β → α.


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In particular, if ∆ = ∅ we have: α ≡ β iff `L α → β and `L β → α. Proposition 3.2.20. If L ∈ {Tm, T4m, T45m, TBm, Dm}. Then, the relation ≡∆ is a congruence w.r.t. connectives into Σ00 = {→, ¬}. Proof. It follows from the fact that L contains classical logic over the signature Σ0 . See for instance (RASIOWA; SIKORSKI, 1963). Notation 3.2.21. If L ∈ {Tm, T4m, T45m, TBm, Dm} and ≡∆ is the congruence defined above. Then, [α]∆ = α/≡∆ = {β ∈ F or(Σ0 ) : α ≡∆ β} is the equivalence class of α ∈ F or(Σ0 ) and F or(Σ0 )/≡∆ = {α/≡∆ : α ∈ F or(Σ0 )} is the set of all equivalence classes. Note that the congruence ≡∆ is well-defined in relation to operations into Σ = {→, ¬} but, the connective is not congruential. See below: 00

Proposition 3.2.22. If L ∈ {Tm, T4m, T45m, TBm, Dm}. Then, the connective is not congruential, that is α ≡∆ β ; α ≡∆ β. Proof. The proof will be divided into two cases, the first to L ∈ {Tm, T4m, T45m, TBm} and the second to L = Dm. So, if L ∈ {Tm, T4m, T45m, TBm}, then we have p ≡∆ ¬p → p but, p 6≡∆ (¬p → p), because there exist a valuation v : F or(Σ0 ) → {tn , tc , f c , f i } for L, such that: • If L ∈ {Tm, TBm}, then v(p) = tc and for v(¬p → p) = tn , we have v(p) ∈ {f i , f c }, but v((¬p → p)) ∈ {tn , tc }; • If L = T4m, then v(p) = tc and for v(¬p → p) = tn , we have v(p) ∈ {f i , f c }, but v((¬p → p)) ∈ {tn }; • If L = T45m, then v(p) = tc and for v(¬p → p) = tn , we have v(p) ∈ {f i }, but v((¬p → p)) ∈ {tn }.

And if L = Dm, we have p ≡∆ ¬p → p but, p 6≡∆ (¬p → p) too. Because there exist a valuation v : F or(Σ0 ) → {T + , T − , C + , C − , F + , F − } for Dm, such that if v(p) = C + and for v(¬p → p) = T + , we have v(p) ∈ {T − , F − , C − }, but v((¬p → p)) ∈ {T + , F + , C + }.


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Through the example, we see that the operator is not well defined because its representative can change. As Carnielli and Coniglio did in (CARNIELLI; CONIGLIO, 2016), to circumvent a similar problem with the operators ◦ and ¬ of the system mbC, we will use the concept of multioperation to dribble the problem, that is, we will define a structure where the range of operations is a set that contains all their possible representatives. Proposition 3.2.23. If L ∈ {Tm, T4m, T45m, TBm, Dm} and let A∆ = hF or(Σ0 )/≡∆ , ∨, ∧, →, 0∆ , 1∆ i be the structure such that [α]∆ ∨ [β]∆ = [¬α → β]∆ [α]∆ ∧ [β]∆ = [¬(α → ¬β)]∆ 0∆ = [¬(α → α)]∆ , 1∆ = [α → α]∆ , Then, A∆ is a Boolean algebra. Proof. The operations ∧ and ∨ are well defined by Proposition 3.2.20. Since L contains classical logic, it is easy to see that A∆ = hF or(Σ0 )/≡∆ , ∨, ∧, →, 0∆ , 1∆ i is a Heyting algebra. For any elements [α]∆ , [β]∆ ∈ F or(Σ0 )/≡∆ , we have [α ∨ (α → β)]∆ = [α]∆ ∨ ([α]∆ → [β]∆ ) = [α]∆ ∨ (¬[α]∆ ∨ [β]∆ ) = [α]∆ ∨ ([¬α]∆ ∨ [β]∆ ) = ([α]∆ ∨ [¬α]∆ ) ∨ [β]∆ = [α ∨ ¬α]∆ ∨ [β]∆ = [1]∆ ∨ [β]∆ = [1]∆ . Then, by Proposition 0.2.42, A∆ is a Boolean algebra. Definition 3.2.24 (Lindenbaum-Tarski swap structure for Tm). Given the Boolean algebra A∆ = hF or(Σ0 )/≡∆ , ∨, ∧, →, 0∆ , 1∆ i, the Lindenbaum-Tarski swap structure for Tm over A∆ is the unique swap structure Tm BA = hBTm A∆ , →, ¬, i, ∆

such that 0 3 BTm A∆ = {([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) ∈ (F or(Σ )/≡∆ ) : [α2 → α1 ]∆ = 1∆ and [α1 ∧ α3 ]∆ = 0∆ }.

Being so, for every ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) and ([β1 ]∆ , [β2 ]∆ , [β3 ]∆ ) in BTm A∆ the multioperations are defined as follows:


81

(i) ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) → ([β1 ]∆ , [β2 ]∆ , [β3 ]∆ ) = {([δ1 ]∆ , [δ2 ]∆ , [δ3 ]∆ ) ∈ BTm A∆ : [δ1 ]∆ = [α1 ]∆ → [β1 ]∆ , [α3 ]∆ ∨ [β2 ]∆ ≤ [δ2 ]∆ ≤ ([α1 ]∆ → [β1 ]∆ ) ∧ ([α2 ]∆ → [β2 ]∆ ) ∧ ([β3 ]∆ → [α3 ]∆ ) and [δ3 ]∆ = [α2 ]∆ ∧ [β3 ]∆ }; (ii) ¬([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) = {([¬α1 ]∆ , [α3 ]∆ , [α2 ]∆ )}; (iii) ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) = {([δ1 ]∆ , [δ2 ]∆ , [δ3 ]∆ ) ∈ BTm A∆ : [δ1 ]∆ = [α2 ]∆ }. Definition 3.2.25 (Lindenbaum-Tarski swap structure for T4m). The LindenbaumTarski swap structure for T4m over A∆ is the unique swap structure for T4m with domain BTm A∆ . T4m Note that, for every ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) in BTm A∆ = BA∆ , the multioperation is given by: ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) = {([δ1 ]∆ , [δ2 ]∆ , [δ3 ]∆ ) ∈ BTm A∆ : [δ1 ]∆ = [α2 ]∆ and [α2 ]∆ ≤ [δ2 ]∆ }.

Definition 3.2.26 (Lindenbaum-Tarski swap structure for T45m). The LindenbaumTarski swap structure for T45m over A∆ is the unique swap structure for T45m with domain BTm A∆ . T45m Note that, for every ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) in BTm , the multioperation A∆ = BA∆ Tm is given by: ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) = {([δ1 ]∆ , [δ2 ]∆ , [δ3 ]∆ ) ∈ BA∆ : [δ1 ]∆ = [α2 ]∆ , [α2 ]∆ ≤ [δ2 ]∆ and [δ3 ]∆ ∨ [δ1 ]∆ = 1∆ }.

Definition 3.2.27 (Lindenbaum-Tarski swap structure for TBm). The LindenbaumTarski swap structure for TBm over A∆ is is the unique swap structure for TBm with domain BTm A∆ . TBm Note that, for every ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) in BTm A∆ = BA∆ , the multioperation is given by: ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) = {([δ1 ]∆ , [δ2 ]∆ , [δ3 ]∆ ) ∈ BTm A∆ : [δ1 ]∆ = [α2 ]∆ and [α1 ]∆ ∨ [δ3 ]∆ = 1∆ }.

Definition 3.2.28 (Lindenbaum-Tarski swap structure for Dm). Given the Boolean algebra A∆ = hF or(Σ0 )/≡∆ , ∨, ∧, →, 0∆ , 1∆ i, the Lindenbaum-Tarski swap structure for Dm over A∆ is the unique swap structure Dm BA = hBDm A∆ , →, ¬, i, ∆

such that 0 3 BDm A∆ = {([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) ∈ (F or(Σ )/≡∆ ) : [α2 ∧ α3 ]∆ = 0∆ }.

Being so, for every ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) and ([β1 ]∆ , [β2 ]∆ , [β3 ]∆ ) in BDm A∆ the multioperations are defined as follows:


82

(i) ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) → ([β1 ]∆ , [β2 ]∆ , [β3 ]∆ ) = {([δ1 ]∆ , [δ2 ]∆ , [δ3 ]∆ ) ∈ BDm A∆ : [δ1 ]∆ = [α1 ]∆ → [β1 ]∆ , [α3 ]∆ ∨ [β2 ]∆ ≤ [δ2 ]∆ ≤ ([α2 ]∆ → [β2 ]∆ ) ∧ ([β3 ]∆ → [α3 ]∆ ) and [δ3 ]∆ = [α2 ]∆ ∧ [β3 ]∆ }; (ii) ¬([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) = {([¬α1 ]∆ , [α3 ]∆ , [α2 ]∆ )}; (iii) ([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) = {([δ1 ]∆ , [δ2 ]∆ , [δ3 ]∆ ) ∈ BDm A∆ : [δ1 ]∆ = [α2 ]∆ }. Definition 3.2.29 (Nmatrix associated to Lindenbaum-Tarski swap structure). Let L ∈ {Tm, T4m, T45m, TBm, Dm} and A∆ be the Boolean algebra L defined as above and let BA be the Lindenbaum-Tarski swap structure over A∆ with ∆ domain BLA∆ . If DBAL = {([α1 ]∆ , [α2 ]∆ , [α3 ]∆ ) ∈ BLA∆ : [α1 ]∆ = 1∆ }, then the non-deterministic ∆ L matrix associated to BA is: ∆ L L M(BA ) = hBA , DBAL i. ∆ ∆ ∆

L L In particular, if ∆ = ∅, M(BA ) = hBA , DBAL i. ∅ ∅ ∅

Definition 3.2.30. If L ∈ {Tm, T4m, T45m, TBm, Dm} and A∆ is defined as in the Proposition 3.2.23, then the relation M(BAL ) is the consequence relation defined over the ∆ L non-deterministic matrix M(BA ) (see Definitions 2.1.4, 2.1.5). ∆ Lemma 3.2.31. If L ∈ {Tm, T4m, T45m, TBm, Dm} and A∆ is defined as in the Proposition 3.2.23, then [α]∆ = 1∆ iff ∆ `L α. Proof. (⇒) By Definition 3.2.29, we have [α]∆ = 1∆ = [p1 ∨ ¬p1 ]∆ . So: 1. ∆ `L p1 ∨ ¬p1 → α 2. ∆ `L p1 ∨ ¬p1 3. ∆ `L α

Definition of ≡∆ CPC MP in 1 and 2

(⇐) Suppose ∆ `L α: 1. 2. 3. 4. 5. 6.

∆ `L ∆ `L ∆ `L ∆ `L ∆ `L ∆ `L

α p1 ∨ ¬p1 α → ((p1 ∨ ¬p1 ) → α) (p1 ∨ ¬p1 ) → α (p1 ∨ ¬p1 ) → (α → (p1 ∨ ¬p1 )) α → (p1 ∨ ¬p1 )

Premise CPC Axiom (Ax1) MP in 1 e 3 Axiom (Ax1) MP in 2 e 5

It shows that ∆ `L (p1 ∨¬p1 ) → α and ∆ `L α → (p1 ∨¬p1 ). So, [α]∆ = [p1 ∨¬p1 ]∆ = 1∆ .


83

Lemma 3.2.32. If L ∈ {Tm, T4m, T45m, TBm, Dm} and A∆ is defined as in the Proposition 3.2.23, then the following conditions are equivalent: (i) [α]∆ ≤ [β]∆ (ii) [α]∆ → [β]∆ = 1∆ (iii) [α → β]∆ = 1∆ (iv) ∆ `L α → β Proof. (i) ⇔ (ii) it is a well known classical result in a Boolean algebra. (ii) ⇔ (iii) it follows from the Proposition 3.2.20. (iii) ⇔ (iv) it follows from the Lemma 3.2.31. Lemma 3.2.33. If A∆ is defined as in the Proposition 3.2.23. The following results hold in the indicated Lindenbaum-Tarski swap structures: L (i) [¬¬α]∆ = [α]∆ in BA , for L ∈ {Tm, T4m, T45m, TBm, Dm}; ∆ L (ii) [α]∆ ≤ [α]∆ in BA , for L ∈ {T4m, T45m}; ∆ T45m (iii) [¬α]∆ ∨ [α]∆ = 1∆ in BA ; ∆ TBm (iv) [α]∆ ∨ [¬α]∆ = 1∆ in BA ; ∆ L (v) [¬(α → β)]∆ = [α]∆ ∧[¬β]∆ in BA , for L ∈ {Tm, T4m, T45m, TBm, Dm}; ∆ L (vi) [¬α]∆ ∨ [β]∆ ≤ [(α → β)]∆ in BA , for L ∈ {Tm, T4m, T45m, TBm, Dm}; ∆ L (vii) [(α → β)]∆ ≤ [α]∆ → [β]∆ in BA , for L ∈ {Tm, T4m, T45m, TBm}; ∆ L (viii) [(α → β)]∆ ≤ [α]∆ → [β]∆ in BA , for L ∈ {Tm, T4m, T45m, TBm, Dm}; ∆ L (ix) [(α → β)]∆ ≤ [¬β]∆ → [¬α]∆ in BA , for L ∈ {Tm, T4m, T45m, TBm, Dm}. ∆

Proof. In this proof we will use the Lemma 3.2.32 to simplify the proof. (i) [¬¬α]∆ = [α]∆ iff ∆ `L ¬¬α → α and ∆ `L α → ¬¬α. We can get these two conditions by applying the axioms (DN 1) and (DN 2). (ii) [α]∆ ≤ [α]∆ iff ∆ `L α → α. But this holds by Axiom (4). (iii) [¬α]∆ ∨ [α]∆ = 1∆ iff [¬α ∨ α]∆ = 1∆ iff [¬¬α → α]∆ = 1∆ iff ∆ `T45m ¬¬α → α. But this holds by Axiom (5).


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(iv) [α]∆ ∨ [¬α]∆ = 1∆ iff [α ∨ ¬α]∆ = 1∆ iff [¬¬α → α]∆ = 1∆ iff ∆ `TBm ¬¬α → α. But this holds by Axiom (B). (v) [¬(α → β)]∆ = [α]∆ ∧ [¬β]∆ iff [¬(α → β)]∆ = [α ∧ ¬β]∆ iff ∆ `L ¬(α → β) → (α ∧ ¬β) and ∆ `L (α ∧ ¬β) → ¬(α → β). See the deduction below: 1. 2. 3. 4. 5.

∆ `L ∆ `L ∆ `L ∆ `L ∆ `L

¬¬(α → β) → (α → ¬¬β) ¬(α → ¬¬β) → ¬¬¬(α → β) ¬(¬α ∨ ¬¬β) → ¬(α → β) (¬¬α ∧ ¬¬¬β) → ¬(α → β) (α ∧ ¬β) → ¬(α → β)

Axiom (k2) CPC in 1 CPC in 2 CPC in 3 CPC in 4

On the other hand: 1. ∆ `L ¬(α → β) → ¬β Axiom (M3) 2. ∆ `L ¬(α → β) → α Axiom (M4) 3. ∆ `L ¬(α → β) → (α ∧ ¬β) CPC in 1 and 2 (vi) [¬α]∆ ∨ [β]∆ ≤ [(α → β)]∆ iff [¬α ∨ β]∆ ≤ [(α → β)]∆ iff ∆ `L (¬α ∨ β) → (α → β). See the deduction below: 1. ∆ `L ¬α → (α → β) 2. ∆ `L β → (α → β) 3. ∆ `L (¬α ∨ β) → (α → β)

Axiom (M1) Axiom (M2) CPC in 1 and 2

(vii) [(α → β)]∆ ≤ [α]∆ → [β]∆ iff [(α → β)]∆ ≤ [α → β]∆ iff ∆ `L (α → β) → (α → β). But this is a consequence of Axiom (T). (viii) [(α → β)]∆ ≤ [α]∆ → [β]∆ iff [(α → β)]∆ ≤ [α → β]∆ iff ∆ `L (α → β) → (α → β). But this follows by Axiom (K). (ix) [(α → β)]∆ ≤ [¬β]∆ → [¬α]∆ iff [(α → β)]∆ ≤ [¬β → ¬α]∆ iff ∆ `L (α → β) → (¬β → ¬α). But this is a consequence of Axiom (K1).

Proposition 3.2.34. If L ∈ {Tm, T4m, T45m, TBm, Dm}, then the function v∆ : L F or(Σ0 ) → BLA∆ , such that v∆ (α) = ([α]∆ , [α]∆ , [¬α]∆ ) is a valuation in M(BA ). ∆ 0 That is, for every α, β ∈ F or(Σ ): (i) v∆ (¬α) ∈ ¬v∆ (α); (ii) v∆ (α) ∈ v∆ (α); (iii) v∆ (α → β) ∈ v∆ (α) → v∆ (β).


Proof.

85

(i) v∆ (¬α) = ([¬α]∆ , [¬α]∆ , [¬¬α]∆ ).

On the other hand, for L ∈ {Tm, T4m, T45m, TBm, Dm}, we have by Proposition 3.2.20: ¬v∆ (α) = ¬([α]∆ , [α]∆ , [¬α]∆ ) = {([¬α]∆ , [¬α]∆ , [α]∆ )} and by Lemma 3.2.33 (i), [¬¬α]∆ = [α]∆ . Thus v∆ (¬α) ∈ ¬v∆ (α); (ii) v∆ (α) = ([α]∆ , [α]∆ , [¬α]∆ ). On the other hand: • For L ∈ {Tm, Dm}, we have: v∆ (α) = ([α]∆ , [α]∆ , [¬α]∆ ) = {([β1 ]∆ , [β2 ]∆ , [β3 ]∆ ) ∈ BLA∆ : [β1 ]∆ = [α]∆ }. Therefore v∆ (α) ∈ v∆ (α); • For L = T4m, we have: v∆ (α) = ([α]∆ , [α]∆ , [¬α]∆ ) = {([β1 ]∆ , [β2 ]∆ , [β3 ]∆ ) ∈ BLA∆ : [β1 ]∆ = [α]∆ and [α]∆ ≤ [β2 ]∆ } and by Lemma 3.2.33 (ii) [α]∆ ≤ [α]∆ . Therefore v∆ (α) ∈ v∆ (α); • For L = T45m, we have: v∆ (α) = ([α]∆ , [α]∆ , [¬α]∆ ) = {([β1 ]∆ , [β2 ]∆ , [β3 ]∆ ) ∈ BLA∆ : [β1 ]∆ = [α]∆ , [α]∆ ≤ [β2 ]∆ and [β3 ]∆ ∨ [β1 ]∆ = 1∆ } and by Lemma 3.2.33 (ii) and (iii) [α]∆ ≤ [α]∆ and [¬α]∆ ∨[α]∆ = 1∆ . Therefore v∆ (α) ∈ v∆ (α); • For L = TBm, we have: v∆ (α) = ([α]∆ , [α]∆ , [¬α]∆ ) = {([β1 ]∆ , [β2 ]∆ , [β3 ]∆ ) ∈ BLA∆ : [β1 ]∆ = [α]∆ and [α]∆ ∨ [β3 ]∆ = 1∆ } and by Lemma 3.2.33 (iv) [α]∆ ∨ [¬α]∆ = 1∆ . Therefore v∆ (α) ∈ v∆ (α); (iii) v∆ (α → β) = ([α → β]∆ , [(α → β)]∆ , [¬(α → β)]∆ ). On the other hand: • For L ∈ {Tm, T4m, T45m, TBm}, we have: v∆ (α) → v∆ (β) = ([α]∆ , [α]∆ , [¬α]∆ ) → ([β]∆ , [β]∆ , [¬β]∆ ) = {([γ1 ]∆ , [γ2 ]∆ , [γ3 ]∆ ) ∈ BLA∆ : [γ1 ]∆ = [α]∆ → [β]∆ , [γ3 ]∆ = [α]∆ ∧ [¬β]∆ and [¬α]∆ ∨ [β]∆ ≤ [γ2 ]∆ ≤ ([α]∆ → [β]∆ ) ∧ ([α]∆ → [β]∆ ) ∧ ([¬β]∆ → [¬α]∆ )}. By Lemma 3.2.33 (v) and (vi), we have [¬(α → β)]∆ = [α]∆ ∧ [¬β]∆ and [¬α]∆ ∨ [β]∆ ≤ [(α → β)]∆ . And by Lemma 3.2.33 (vii), (viii) and (ix), we have [(α → β)]∆ ≤ ([α]∆ → [β]∆ ) ∧ ([α]∆ → [β]∆ ) ∧ ([¬β]∆ → [¬α]∆ ). Therefore v∆ (α → β) ∈ v∆ (α) → v∆ (β). • For L = Dm, we have: v∆ (α) → v∆ (β) =


86

([α]∆ , [α]∆ , [¬α]∆ ) → ([β]∆ , [β]∆ , [¬β]∆ ) = {([γ1 ]∆ , [γ2 ]∆ , [γ3 ]∆ ) ∈ BLA∆ : [γ1 ]∆ = [α]∆ → [β]∆ , [γ3 ]∆ = [α]∆ ∧ [¬β]∆ and [¬α]∆ ∨ [β]∆ ≤ [γ2 ]∆ ≤ ([α]∆ → [β]∆ ) ∧ ([¬β]∆ → [¬α]∆ )}. By Lemma 3.2.33 (v) and (vi), we have [¬(α → β)]∆ = [α]∆ ∧ [¬β]∆ and [¬α]∆ ∨ [β]∆ ≤ [(α → β)]∆ . And by Lemma 3.2.33 (viii) and (ix), we have [(α → β)]∆ ≤ ([α]∆ → [β]∆ ) ∧ ([¬β]∆ → [¬α]∆ ). Therefore v∆ (α → β) ∈ v∆ (α) → v∆ (β).

Proposition 3.2.35. If L ∈ {Tm, T4m, T45m, TBm, Dm}, then the function v∆ : F or(Σ0 ) → BLA∆ , such that v∆ (α) = ([α]∆ , [α]∆ , [¬α]∆ ) is a canonical valuation in L M(BA ), that is v∆ (α) ∈ DBAL iff ∆ `L α. ∆ ∆

Proof. Suppose v∆ (α) = ([α]∆ , [α]∆ , [¬α]∆ ) ∈ DBAL , by Definition 3.2.29, we have ∆ L [α]∆ = 1∆ . So, by Proposition 3.2.34, v∆ (α) is a valuation in M(BA ) and by Lemma 3.2.31, ∆ v∆ (α) ∈ DBAL iff ∆ `L α. ∆

Corollary 3.2.36. For L ∈ {Tm, T4m, T45m, TBm, Dm}, `L α iff `L (β ∨¬β) → α. The canonical valuation allows to prove immediately the completeness of L ∈ {Tm, T4m, T45m, TBm, Dm} w.r.t. swap structures: Theorem 3.2.37. (Completeness) For L ∈ {Tm, T4m, T45m, TBm, Dm} and for every ∆ ∪ {α} ⊆ F or(Σ0 ), ∆ |=M at(KL ) α ⇒ ∆ `L α. Proof. Suppose that ∆ 6`L α. Then by Proposition 3.2.35, there exist a valuation v∆ over L a Lindenbaum-Tarski swap structure BA such that v∆ (α) 6∈ DBAL . ∆ ∆

If β ∈ ∆, then ∆ `L β and applying again the Proposition 3.2.35, we have v∆ (β) ∈ DBAL . Thus, v∆ [∆] ⊆ DBAL but v∆ (α) 6∈ DBAL . From this, ∆ 6|=M at(KL ) α. ∆

∆

∆

Corollary 3.2.38. For L ∈ {Tm, T4m, T45m, TBm, Dm}, M(BAL ) α iff M(BAL ) ∆ ∆ (β ∨ ¬β) → α. Proposition 3.2.39. For L ∈ {Tm, T4m, T45m, TBm, Dm} and for every ∆ ∪ {α} ∪ {β} ⊆ F or(Σ0 ), we have ∆, α M(BAL ) β iff ∆ M(BAL ) α → β. ∆

Proof. ∆, α M(BAL

∆

)

∆

β ⇔ ∆, α `L β ⇔ ∆ `L α → β ⇔ ∆ M(BAL

∆

)

α → β.


87

Proposition 3.2.40. For L ∈ {Tm, T4m, T45m, TBm, Dm} and for every ∆ ∪ {α} ⊆ F or(Σ0 ), we have ∆ M(BAL ) α iff there exist ∆0 finite ⊆ ∆, such that ∆0 M(BAL ) ∆ ∆ α. Proof. ∆ M(BAL ) α ⇔ ∆ `L α ⇔ there exist ∆0 finite ⊆ ∆, such that ∆0 `L α ⇔ there ∆ exist ∆0 finite ⊆ ∆, such that ∆ M(BAL ) α. ∆

Remark 3.2.41. In the proof of the next proposition we are assuming that the infimum of the empty set is a tautology. Proposition 3.2.42. For L ∈ {Tm, T4m, T45m, TBm, Dm}, let ∆0 be finite. For V every ∆0 ∪ {α} ⊆ F or(Σ0 ), we have ∆0 M(BAL ) α iff M(BAL ) ∆0 → α. ∆

Proof.

∆

• If ∆0 = ∅:

M(BAL

∆

α ⇔(3.2.38) M(BAL

)

∆

)

(β ∨ ¬β) → α ⇔ M(BAL

∆

)

V

∆0 → α;

• If ∆0 6= ∅, let ∆0 = {γ1 , γ2 , . . . , γn−1 , γn } : ∆0 M(BAL

∆

)

α⇔

γ1 , γ2 , . . . , γn−1 , γn M(BAL γ1 , γ2 , . . . , γn−1 M(BAL

∆

γ1 , γ2 , . . . , γn−2 M(BAL

∆

M(BAL

∆

M(BAL

∆

M(BAL

∆

∆

)

α ⇔(3.2.39)

)

γn → α ⇔(3.2.39)

)

γn−1 → (γn → α) ⇔(3.2.39)

)

γ1 → (γ2 → (. . . → (γn−1 → (γn → α)) . . .)) ⇔

)

(γ1 ∧ γ2 ∧ . . . ∧ γn−1 ∧ γn ) → α ⇔

)

V

∆0 → α.

Theorem 3.2.43. For L ∈ {Tm, T4m, T45m, TBm, Dm} and for every ∆ ∪ {α} ⊆ F or(Σ0 ), are equivalent: 1. ∆ `L α; 2. ∆ M at(KL ) α; 3. ∆ M(BAL

∆

)

α;

4. ∆ M(BAL ) α; 2

5. ∆ M(BL

A∅ )

α.

Proof. (1) ⇔ (2) By Theorems 3.2.18 and 3.2.37;


88

(1) ⇔ (3) By Theorem 3.2.18 and the proof of Theorem 3.2.37; (1) ⇔ (4) By the soundness and completeness theorems proved by (CONIGLIO; L CERRO; PERON, 2016) and by observing that BA are the finite-valued non2 deterministic matrices presented there; (1) ⇔ (5) ∆ M(BL

A∅ )

α ⇔(3.2.40)

There exist ∆0 finite ⊆ ∆, such that ∆0 M(BL

A∅ )

There exist ∆0 finite ⊆ ∆, such that M(BL

A∅ )

There exist ∆0 finite ⊆ ∆, such that `L

V

V

α ⇔(3.2.42)

∆0 → α ⇔

∆0 → α ⇔5

There exist ∆0 finite ⊆ ∆, such that ∆0 `L α ⇔6 ∆ `L α.

Corollary 3.2.44. For L ∈ {Tm, T4m, T45m, TBm, Dm} and for every α ∈ F or(Σ0 ), are equivalent: 1. `L α; 2. M at(KL ) α; 3. M(BAL

∆

)

α;

4. M(BAL ) α. 2

5 6

By Deduction Theorem and properties. By compactness.

89

4 An algebraic study of LFIs by means of swap structures In this chapter, we will present an algebraic study of LFIs theory using swap structures and the properties and definitions of multialgebras introduced in previous chapters. For this, we will use definitions and properties of multialgebras and the development in the category theory started in the Chapter 2, and we will apply the method of Lindenbaum-Tarski swap structures, introduced in the Chapter 3 to obtain completeness theorems w.r.t Hilbert-style version of these logical systems. This results are based on the pre-print (CONIGLIO; ORELLANO; GOLZIO, 2016).

The C-systems of da Costa (COSTA, 1963; COSTA, 1974) motivated the emergence of several paraconsistent logics in the last years. That was the case of Logics of Formal Inconsistency (LFIs, for short) introduced by W. Carnielli and J. Marcos in (CARNIELLI; MARCOS, 2002), as a generalization of da Costa systems Cn . In its simplest form, the LFIs have a non-explosive negation ¬, as well as a (primitive or derived) consistency connective ◦ which allows to recover the explosion law in a controlled way. Definition 4.0.1. Let L = hΘ, ì be a Tarskian, finitary and structural logic defined over a propositional signature Θ, which contains a negation ¬, and let ◦ be a (primitive or defined) unary connective. Then, L is said to be a Logic of Formal Inconsistency with respect to ¬ and ◦ if the following holds: (i) α, ¬α 0 β for some α and β; (ii) there are two formulas ϕ and ψ such that (ii.a) ◦ϕ, ϕ 0 ψ; (ii.b) ◦ϕ, ¬ϕ 0 ψ; (iii) ◦α, α, ¬α ` β for every α and β. Condition (ii) of the definition of LFIs is required in order to satisfy condition (iii) in a non-trivial way. The hierarchy of LFIs studied in (CARNIELLI; CONIGLIO; MARCOS, 2007) and (CARNIELLI; CONIGLIO, 2016) starts from a logic called mbC, which extends classical positive logic CPL+ by adding a negation ¬ and an unary consistency operator ◦ satisfying minimal requirements in order to define an LFI.

Chapter 4. An algebraic study of LFIs by means of swap structures

90

As it is well-known, several logics in the hierarchy of the LFIs cannot be semantically characterized by a single finite matrix. Moreover, they lie outside the scope of the usual techniques of algebraization of logics such as Blok and Pigozzi’s method. Several alternative semantical tools were introduced in the literature in order to deal with such systems: (non-truth-functional) bivaluations, possible-translations semantics, and non-deterministic matrices (or Nmatrices), obtaining so, decision procedures for these logics. However, the problem of finding an algebraic counterpart for this kind of logic, in a sense to be determined, remains open. So, we use the concepts of multialgebra, swap structure and non-deterministic matrix to get some results towards the possibility of defining an algebraic theory of swap structures semantics. As a first step, we will concentrate our efforts on the algebraic theory of KmbC , the class of swap structures for the logic mbC and we will prove that the class KmbC is closed under sub-swap-structures and products, but it is not closed under homomorphic images, hence it is not a variety in the usual sense. We will also show that is possible to give a representation theorem for KmbC which is analogous to the Birkhoff’s theorem in traditional algebraic logics (see Theorem 4.2.15). As a consequence of this, the class KmbC is generated by the structure with five elements, which is constructed over the 2-element Boolean algebra. Such structure is precisely Avron’s 5-valued characteristic non-deterministic matrix for mbC. And finally, we will prove that, under the present approach, the classes of swap structures for the axiomatic extensions of mbC found in (CARNIELLI; CONIGLIO, 2016) are subclasses of KmbC . They are obtained by requiring that its elements satisfy precisely the additional axioms which define the corresponding logic. Analogous representation theorems for each class of multialgebras will be found and this allow a modular treatment of the algebraic theory of swap structures, as happens in the traditional algebraic setting.

From now on, the following signatures will be considered: Σ+ = {∧, ∨, →}; ΣBA = {∧, ∨, →, 0, 1} and Σ = {∧, ∨, →, ¬, ◦}. Remark 4.0.2. Although we use the same symbols ¬ and → in Σ = {∧, ∨, →, ¬, ◦} and Σ0 = {¬, →, } (recall Chapter 3), in each signature these symbols have a different interpretation. So the approach given to operators ¬ and → in the previous chapter is different from that is done in this chapter.


91

Definition 4.0.3 (Classical Positive Logic). The classical positive logic CPL+ is defined over the language F or(Σ+ ) by the following Hilbert-style deductive system:

Axiom schemas: α → (β → α)

(Ax1)

(α → (β → γ)) → ((α → β) → (α → γ)) α → (β → (α ∧ β))

(Ax3)

(α ∧ β) → α

(Ax4)

(α ∧ β) → β

(Ax5)

α → (α ∨ β)

(Ax6)

β → (α ∨ β)

(Ax7)

(α → γ) → ((β → γ) → ((α ∨ β) → γ)) (α → β) ∨ α

(Ax2)

(Ax8)

(Ax9)

Inference rule modus ponens: α, α → β β

(M P )

Definition 4.0.4. The logic mbC, defined over signature Σ, is obtained from CPL+ by adding the following axiom schemas: α ∨ ¬α

(Ax10)

◦α → (α → (¬α → β))

(bc1)

For convenience, a (purely linguistic) expansion of CPL+ over signature Σ will be considered from now on, besides CPL+ itself. This logic, denoted by CPL+ e , is nothing more than CPL+ defined over Σ, that is, by adding ¬ and ◦ without any axioms or rules for them. The simplest semantical characterization of mbC is given in terms of bivaluations or mbC-valuations (see (CARNIELLI; CONIGLIO; MARCOS, 2007)).


92

Definition 4.0.5 (Valuations for mbC). A function v : F or(Σ) → {0, 1} is an mbCvaluation, if it satisfies the following conditions: v(α ∧ β) = 1 ⇔ v(α) = 1 and v(β) = 1

(vAnd)

v(α ∨ β) = 1 ⇔ v(α) = 1 or v(β) = 1

(vOr)

v(α → β) = 1 ⇔ v(α) = 0 or v(β) = 1 v(α) = 0 ⇒ v(¬α) = 1

(vImp)

(vN eg)

v(◦α) = 1 ⇒ v(α) = 0 or v(¬α) = 0

(vCon)

Definition 4.0.6 (Semantical consequence relation). For every Γ ∪ {α} ⊆ F or(Σ), Γ mbC α iff, for every mbC-valuation v, if v(γ) = 1 for every γ ∈ Γ, then v(α) = 1. Proposition 4.0.7 (Soundness and completeness w.r.t. valuations). For every Γ ∪ {α} ⊆ F or(Σ), Γ `mbC α iff Γ mbC α. Proof. In (CARNIELLI; CONIGLIO; MARCOS, 2007, p. 38 and 40).

4.1 Swap structures for CPL+ e In (CARNIELLI; CONIGLIO, 2016) it was introduced the notion of swap structures for mbC, as well as for some axiomatic extensions of it. In this section, these structures will be reintroduced in a slightly more general form, in order to define a hierarchy of classes of multialgebras associated to the corresponding hierarchy of logics. This is in line with the traditional approach of algebraic logic, in which hierarchies of classes of algebraic models are associated to hierachies of logics. Since mbC is an axiomatic extension of CPL+ e , it is natural to begin with swap structures for the latter logic. The algebraic semantics for CPL+ is given by classical implicative lattices (recall Definition 0.2.45). Namely, Γ `CPL+ α iff, for every classical implicative lattice A and for every homomorphism v : F or(Σ+ ) → A, if v(γ) = 1 for every γ ∈ Γ then v(α) = 1. Now a semantics of swap structures over a given classical implicative lattice A will be introduced for CPL+ e . The idea is that a triple (c1 , c2 , c3 ) in such structure represents a (composite) truth-value in which c1 represents the truth-value of a formula α, while c2 and c3 represent a possible truth-value for ¬α and ◦α, respectively. Observe the analogy with the semantics approach to modal logics given in the previous chapter. Let A = hA, ∧, ∨, →i be classical implicative lattice and let πj : A3 → A be the canonical projections, for 1 ≤ j ≤ 3.


93

Definition 4.1.1. Let A be a classical implicative lattice with domain A. The universe CPL+ e of swap structures for CPL+ = A3 . e over A is the set BA Definition 4.1.2. Let A = hA, ∧, ∨, →i be a classical implicative lattice and let B ⊆ + e BCPL . A swap structure for CPL+ A e over A is any multialgebra B = hB, ∧, ∨, →, ¬, ◦i over Σ such that the multioperations satisfy the following, for every (a1 , a2 , a3 ) and (b1 , b2 , b3 ) in B: (i) ∅ = 6 (a1 , a2 , a3 )#(b1 , b2 , b3 ) ⊆ {(c1 , c2 , c3 ) ∈ B : c1 = a1 #b1 }, for # ∈ {∧, ∨, →}; (ii) ∅ = 6 ¬(a1 , a2 , a3 ) ⊆ {(c1 , c2 , c3 ) ∈ B : c1 = a2 }; (iii) ∅ = 6 ◦(a1 , a2 , a3 ) ⊆ {(c1 , c2 , c3 ) ∈ B : c1 = a3 }. +

CPLe For every classical implicative lattice A, there is a unique swap structure BA + e with domain BCPL = A3 such that, for every (a1 , a2 , a3 ) and (b1 , b2 , b3 ) in A3 : A

(i) (a1 , a2 , a3 )#(b1 , b2 , b3 ) = {(c1 , c2 , c3 ) ∈ A3 : c1 = a1 #b1 }, for # ∈ {∧, ∨, →}; (ii) ¬(a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ A3 : c1 = a2 }; (iii) ◦(a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ A3 : c1 = a3 }. +

CPLe Remark 4.1.3. Observe that, if B is a swap structure for CPL+ e over A, then B ⊆ BA in the sense of Definition 2.2.1 (as submultialgebra).

Proposition 4.1.4. Let A be a classical implicative lattice with domain A and let B a swap structure for CPL+ e over A with domain B, then π1 [B] (with the operations naturally induced by composing π1 with the multioperations of B) is a classical implicative lattice included in A as subalgebra. Proof. By definition of multioperations in B, π1 [(c1 , c2 , c3 )#(d1 , d2 , d3 )] = {c1 #d1 } for # ∈ {∧, ∨, →}. So, π1 [B] is closed under the operations ∧, ∨ and → of A: if a1 , b1 ∈ π1 [B], then a1 #b1 ∈ π1 [B] for # ∈ {∧, ∨, →}. Now, let c = (c1 , c2 , c3 ) ∈ B. By definition of multioperations in B, π1 [c → c] = {c1 → c1 } = {1} and so 1 ∈ π1 [B]. Now, since π1 [(a1 , a2 , a3 ) → (b1 , b2 , b3 )] = {a1 → b1 } and A is a classical implicative lattice, then a1 → W b1 = {c1 ∈ A : a1 ∧c1 ≤ b1 }, for every a1 , b1 ∈ A. Additionally, let (a1 , a2 , a3 ), (b1 , b2 , b3 ) ∈ B, π1 [(a1 , a2 , a3 ) ∨ ((a1 , a2 , a3 ) → (b1 , b2 , b3 ))] = {a1 ∨ (a1 → b1 )} = {1}. Therefore, π1 [B] is a classical implicative lattice.


94

As in the previous chapter, elements of a swap structure for CPL+ e are called + snapshots for CPLe (or simply snapshots). Since no axioms or rules are given in CPL+ e for the new connectives ¬ and ◦, the multioperations associated to it in a swap structure just put in evidence (or ‘swap’) its value on the first coordinate, leaving free the value of the other cordinates, obtaining so a (non-empty) set of snapshots. As we shall see in the next sections, when axioms are given for ¬ and ◦, the multioperations (and the domain of the swap structures themselves) must be restricted.

Now, we will use the notions of semantics associated to non-deterministic matrices introduced by Avron and Lev (see Definitions 2.1.4 and 2.1.5). Let K+ be the class of swap structures for CPL+ e. CPL+ e

As it was done in (CARNIELLI; CONIGLIO, 2016, Chapter 6) with several LFIs, and as it was done in the previous chapter, it is easy to see that each B ∈ K+ CPL+ e induces naturally a non-deterministic matrix such that the class of such non-deterministic matrices semantically characterizes CPL+ e . More precisely: Definition 4.1.5. For each B ∈ K+ , let DB = {(c1 , c2 , c3 ) ∈ |B| : c1 = 1}. CPL+ e

1. The non-deterministic matrix associated to B is M(B) = (B, DB ); n

o

2. M at(K+ ) = M(B) : B ∈ K+ is the class of the non-deterministic matrices CPL+ CPL+ e e associated to B. In this particular case, Definition 2.1.5 assumes the following form: Definition 4.1.6. Let B ∈ K+ and M(B) as above. A valuation over M(B) is a CPL+ e function v : F or(Σ) → |B| such that, for every α, β ∈ F or(Σ): 1. v(α#β) ∈ v(α)#v(β), for every # ∈ {∧, ∨, →}; 2. v(¬α) ∈ ¬v(α); 3. v(◦α) ∈ ◦v(α). Definition 4.1.7. Let Γ ∪ {α} ⊆ F or(Σ) be a set of formulas of CPL+ e . We say that α + is a consequence of Γ in the class M at(KCPL+ ) of non-deterministic matrices, denoted e by Γ |=M at(K+ ) α, if Γ |=M α for every M ∈ M at(K+ ). In particular, α is valid in CPL+ CPL+ e

M at(K+ ), CPL+ e

e

denoted by |=M at(K+

) CPL+ e

α, if it is valid in every M ∈ M at(K+ ). CPL+ e

In the following theorem, in order to prove completeness, we will apply, once again the method of the Lindenbaum-Tarski swap structure already used in the previous chapter for modal systems.


95

Theorem 4.1.8 (Adequacy of CPL+ e w.r.t. swap structures). Let Γ ∪ {α} ⊆ F or(Σ) be + a set of formulas of CPLe . Then: Γ `CPL+e α iff Γ |=M at(K+ ) α. CPL+ e

Proof. ‘Only if’ part (Soundness): It follows by the definition of semantics of swap structures, and by the fact that CPL+ is sound w.r.t. classical implicative lattices. +

CPLe Indeed, if v is a valuation over B ⊆ BA then h = π1 ◦ v : F or(Σ) → A is a Σ+ -homomorphism such that h(γ) = 1 iff v(γ) ∈ DB : If # ∈ {∧, ∨, →}, v(ϕ) # v(ψ) = {(c1 , c2 , c3 ) ∈ |B| : c1 = π1 ◦ v(ϕ) # π1 ◦ v(ψ)}, so π1 [v(ϕ) # v(ψ)] = {π1 ◦ v(ϕ) # π1 ◦ v(ψ)}. By Definition 4.1.6, we have v(ϕ # ψ) ∈ v(ϕ) # v(ψ), thus π1 ◦ v(ϕ # ψ) = π1 (v(ϕ) # v(ψ)) = π1 ◦ v(ϕ) # π1 ◦ v(ψ) and therefore h(ϕ # ψ) = h(ϕ) # h(ψ). Finally, by Definition 4.1.5 we have h(γ) = 1 iff π1 ◦ v(γ) = 1 iff v(γ) ∈ DB .

If α is an (instance of an) axiom of CPL+ e then h(α) = 1 and so v(α) ∈ DB : See, for instance, if α is the (instance of the) Axioma (Ax1): h(γ → (β → γ)) = π1 ◦ v(γ → (β → γ)) = π1 ◦ v(γ) → (π1 ◦ v(β) → π1 ◦ v(γ)) = 1. Since h(ϕ # ψ) = h(ϕ) # h(ψ) for # ∈ {∧, ∨, →}, the proof for the remaining axioms of CPL+ e is quite similar. On the other hand, if α follows from two previous formulas by (M P ), that is: if v(γ) ∈ DB and v(γ → α) ∈ DB then v(α) ∈ DB , because by hypothesis π1 ◦ v(γ) = 1 and π1 ◦ v(γ → α) = π1 ◦ v(γ) → π1 ◦ v(α) = 1, so, clearly π1 ◦ v(α) = 1. Γ |=M at(K+

Thus, by induction on the length of a derivation of α from Γ, Γ `CPL+e α implies ) α.

CPL+ e

‘If’ part (Completeness): Suppose that Γ 0CPL+e α. Define in F or(Σ) the following relation: ϕ ≡Γ ψ iff Γ `CPL+e ϕ → ψ and Γ `CPL+e ψ → ϕ. It is clearly an def equivalence relation. Let AΓ = F or(Σ)/≡Γ be the quotient set, and define over AΓ the def following operations: [ϕ]Γ # [ψ]Γ = [ϕ # ψ]Γ , for # ∈ {∧, ∨, →} (here, [ϕ]Γ denotes the equivalence class of ϕ w.r.t. ≡Γ ). These operations are clearly1 well-defined, and so they induce a structure of classical implicative lattice over the set AΓ , that is [ϕ]Γ ≤ [ψ]Γ see Γ `CPL+e ϕ → ψ is a partial ordering and [ϕ]Γ ∧ [ψ]Γ ≤ [θ]Γ see [ψ]Γ ≤ [ϕ → θ]Γ . Let AΓ CPL+ e be the obtained classical implicative lattice, and let BA be the corresponding swap Γ + def + CPLe CPLe structure. Let MΓ = M(BAΓ ), and consider the mapping vΓ : F or(Σ) → (AΓ )3 given by vΓ (ϕ) = ([ϕ]Γ , [¬ϕ]Γ , [◦ϕ]Γ ). Then vΓ is a valuation over the non-deterministic + e matrix MCPL such that, for every ϕ, vΓ (ϕ) ∈ D CPL+e iff Γ `CPL+e ϕ, see the proof below Γ BA Γ : For every ϕ, ψ ∈ F or(Σ), we have: • vΓ (ϕ # ψ) = ([ϕ # ψ]Γ , [¬(ϕ # ψ)]Γ , [◦(ϕ # ψ)]Γ ) and 1

Using the axioms (Ax1) − (Ax8) of the Definition 4.0.3.


96

vΓ (ϕ) # vΓ (ψ) = ([ϕ]Γ , [¬ϕ]Γ , [◦ϕ]Γ ) # ([ψ]Γ , [¬ψ]Γ , [◦ψ]Γ ) = {(c1 , c2 , c3 ) ∈ (AΓ )3 : def c1 = [ϕ]Γ # [ψ]Γ }, for # ∈ {∧, ∨, →}. Since [ϕ]Γ # [ψ]Γ = [ϕ # ψ]Γ , for # ∈ {∧, ∨, →}, then vΓ (ϕ # ψ) ∈ vΓ (ϕ) # vΓ (ψ). • vΓ (¬ϕ) = ([¬ϕ]Γ , [¬¬ϕ]Γ , [◦¬ϕ]Γ ) and ¬(vΓ (ϕ)) = ¬([ϕ]Γ , [¬ϕ]Γ , [◦ϕ]Γ ) = {(c1 , c2 , c3 ) ∈ (AΓ )3 : c1 = [¬ϕ]Γ }. So vΓ (¬ϕ) ∈ ¬(vΓ (ϕ)). • vΓ (◦ϕ) = ([◦ϕ]Γ , [¬◦ϕ]Γ , [◦◦ϕ]Γ ) and ◦(vΓ (ϕ)) = ◦([ϕ]Γ , [¬ϕ]Γ , [◦ϕ]Γ ) = {(c1 , c2 , c3 ) ∈ (AΓ )3 : c1 = [◦ϕ]Γ }. So vΓ (◦ϕ) ∈ ◦(vΓ (ϕ)). +

e Thus vΓ is a valuation over MCPL . Γ

Fact: vΓ (ϕ) ∈ D

CPL+ e

BA

D

CPL+ e Γ

BA

iff Γ `CPL+e ϕ. Indeed, suppose that vΓ (ϕ) = ([ϕ]Γ , [¬ϕ]Γ , [◦ϕ]Γ ) ∈

Γ

. By definition [ϕ]Γ = 1Γ = [p1 ∨ ¬p1 ]Γ . So: 1. Γ `CPL+e (p1 ∨ ¬p1 ) → ϕ Definition of ≡Γ 2. Γ `CPL+e p1 ∨ ¬p1 Ax10 3. Γ `CPL+e ϕ MP in 1 and 2 Conversely, suppose that Γ `CPL+e ϕ: 1. 2. 3. 4. 5. 6.

Γ `CPL+e Γ `CPL+e Γ `CPL+e Γ `CPL+e Γ `CPL+e Γ `CPL+e

ϕ p1 ∨ ¬p1 ϕ → ((p1 ∨ ¬p1 ) → ϕ) (p1 ∨ ¬p1 ) → ϕ (p1 ∨ ¬p1 ) → (ϕ → (p1 ∨ ¬p1 )) ϕ → (p1 ∨ ¬p1 )

hypothesis Ax10 Ax1 MP in 1 and 3 Ax1 MP in 2 and 5

So Γ `CPL+e (p1 ∨ ¬p1 ) → ϕ and Γ `CPL+e ϕ → (p1 ∨ ¬p1 ). Thus, [ϕ]Γ = [p1 ∨ ¬p1 ]Γ = 1Γ and therefore, vΓ (ϕ) = ([ϕ]Γ , [¬ϕ]Γ , [◦ϕ]Γ ) ∈ D CPL+e . This proves the BΓ

fact.

From the fact above, for every γ ∈ Γ we have that Γ `CPL+e γ and therefore vΓ (γ) ∈ D CPL+e . But since Γ 0CPL+e α, then vΓ (α) 6∈ D CPL+e . Hence, Γ 6|=M at(K+ ) α. BA

Γ

BA

Γ

CPL+ e

The previous result can be improved a bit, by considering swap structures defined over Boolean algebras instead of classical implicative lattices. The reason to do this is that, when considering LFIs, a bottom element arises, and so each classical implicative lattice obtained as in the proof of Theorem 4.1.8 becomes a Boolean algebra. In the sequel, it will be proven that a given classical implicative lattice A can be extended to a Boolean algebra A∗ such that the non-deterministic matrices induced by both algebras validate the sames formulas.


97 def

Definition 4.1.9. Let A = hA, ∧, ∨, →i be a classical implicative lattice, and let A∗ = A× {0, 1}. Consider the operations ∧, ∨ and → defined over A∗ as follows, for every a, b ∈ A: (a, 1)#(b, 1) = (a#b, 1), for # ∈ {∧, ∨, →}; (a, 1) ∧ (b, 0) = (b, 0) ∧ (a, 1) = (a → b, 0); (a, 0) ∧ (b, 0) = (a ∨ b, 0); (a, 1) ∨ (b, 0) = (b, 0) ∨ (a, 1) = (b → a, 1); (a, 0) ∨ (b, 0) = (a ∧ b, 0); (a, 1) → (b, 0) = (a ∧ b, 0); (a, 0) → (b, 1) = (a ∨ b, 1); (a, 0) → (b, 0) = (b → a, 1). Remark 4.1.10. The intuitive reading of (a, 1) is “a”, moreover (a, 0) denotes its Boolean complement ∼a.2 Proposition 4.1.11. The structure A∗ = hA∗ , ∧, ∨, →, 0∗ , 1∗ i where the binary operators {∧, ∨, →} are defined as in Definition 4.1.9, is a non-trivial Boolean algebra (that is, def def 1∗ 6= 0∗ ) such that 0∗ = (1, 0) and 1∗ = (1, 1). Proof. Taking into account that, for any a ∈ A, the pairs (a, 1) and (a, 0) can be considered in A∗ as representing uniquely a and its Boolean complement ∼a, respectively, then we need to prove that hA∗ , ∧, ∨i is a distributive lattice. See the proof of one side of distributivity: ((a, 1) ∧ (b, 1)) ∨ (c, 1) = (a ∧ b, 1) ∨ (c, 1) = ((a ∧ b) ∨ c, 1). Since A is a classical implicative lattice we have ((a ∧ b) ∨ c, 1) = ((a ∨ b) ∧ (a ∨ c), 1) = (a ∨ b, 1) ∧ (a ∨ c, 1) = ((a, 1)∨(b, 1))∧((a, 1)∨(c, 1)). And see the proof that ((a → b) → c, 1) = (a → b, 0)∨(c, 1): ((a → b) → c, 1) = (∼(∼a ∨ b) ∨ c, 1) = ((a ∧ ∼b) ∨ c, 1) = ((a ∨ c) ∧ (∼b ∨ c), 1) = ((a, 1) ∨ (c, 1), 1) ∧ ((b, 0) ∨ (c, 1), 1) = ((a, 1) ∧ (b, 0)) ∨ (c, 1) = (a → b, 0) ∨ (c, 1). The proof of the other items is quite similar. Now, we have (a, 1) ∧ (1, 0) = (a → 1, 0) = (1, 0) (then 0∗ is the bottom element), (a, 1) ∨ (1, 1) = (a ∨ 1, 1) = (1, 1) (then 1∗ is the top element), (a, 1) ∨ (a, 0) = (a → a, 1) = (1, 1) and (a, 1) ∧ (a, 0) = (a → a, 0) = (1, 0) (then (a, 0) is the Boolean complement of (a, 1)). Proposition 4.1.12. Given a classical implicative lattice A, let A∗ as in Proposition 4.1.11. (1) Let i∗ : A → A∗ be the mapping given by i∗ (a) = (a, 1), for every a ∈ A. Then i∗ is a monomorphism of classical implicative lattices. 2

In this chapter we will use the symbol ∼ instead of ¬ to denote the Boolean complement, different from what we did in the previous chapter, because the operator ¬ in the LFIs is not classic as in modal logics.


98

(2) The pair (A∗ , i∗ ) has the following universal property: if A0 is a non-trivial Boolean algebra and h : A → A0 is a homomorphism of classical implicative lattices then there exists a unique homomorphism of Boolean algebras h∗ : A∗ → A0 such that h = h∗ ◦ i∗ . That is, the diagram below commutes. A

i∗

/ A∗

h

& 0 A

h∗

Proof. (1) It is immediate from the definition of A∗ that i∗ is a homomorphism of classical implicative lattices. Suppose that there are two homomorphisms of classical implicative lattices f : A00 → A and g : A00 → A such that i∗ ◦ f = i∗ ◦ g. By definition, for every a ∈ |A00 |, i∗ ◦ f (a) = i∗ (f (a)) = (f (a), 1) and i∗ ◦ g(a) = i∗ (g(a)) = (g(a), 1). But, (f (a), 1) = (g(a), 1) iff f (a) = g(a), for every a ∈ |A00 |. Therefore, f = g. (2) Let h∗ (a, 1) = h(a) and h∗ (a, 0) = ∼h(a) for every a ∈ A, where ∼ denotes the Boolean complement in A0 . Clearly, h∗ is a homomorphism of Boolean algebras. Moreover, for every a ∈ A, h∗ ◦ i∗ (a) = h∗ (i∗ (a)) = h∗ (a, 1) = h(a), thus h = h∗ ◦ i∗ . b ∗ such that h = h b ∗ ◦ i∗ . To prove the uniqueness, suppose that there exist another h b ∗ ◦ i∗ (a) = h b ∗ (i∗ (a)) = h b ∗ (a, 1) and We have, for every a ∈ A, h∗ (a, 1) = h(a) = h b ∗ ◦i∗ (a)) = ∼(h b ∗ (i∗ (a))) = ∼(h b ∗ (a, 1)). Since h b ∗ is a homomorphism h∗ (a, 0) = ∼h(a) = ∼(h b ∗ (a, 1)) = h b ∗ (∼(a, 1)) = h b ∗ (a, 0), so h b ∗ = h∗ . of Boolean algebras, ∼(h Now, let KCPL+e be the subclass of K+ formed by swap structures B for CPL+ e + CPLe over a classical implicative lattice A such that A is in fact a Boolean algebra (that is, A has a bottom element 0). Let M at(KCPL+e ) be the corresponding class of non-deterministic matrices. Theorem 4.1.13 (Adequacy of CPL+ e w.r.t. swap structures). Let Γ ∪ {α} ⊆ F or(Σ) be a set of formulas of CPL+ α iff Γ |=M at(KCPL+ ) α. e . Then: Γ `CPL+ e e

Proof. ‘Only if’ part (Soundness): It follows from Theorem 4.1.8, since KCPL+e ⊆ K+ . CPL+ e

‘If’ part (Completeness): Suppose that Γ 0CPL+e ϕ, and consider the classical implicative lattice AΓ defined as in the proof of Theorem 4.1.8. Let (AΓ )∗ be the Boolean algebra CPL+ e induced by AΓ as in Definition 4.1.9 and Proposition 4.1.11. Let B(A ∗ be the corresponding Γ)

+

∗

def

+

CPLe e swap structure in KCPL+e . Let MCPL = M(B(A ∗ ), and consider now a mapping Γ Γ) ∗ ∗ 3 ∗ vΓ : F or(Σ) → ((AΓ ) ) given by vΓ (ϕ) = (([ϕ]Γ , 1), ([¬ϕ]Γ , 1), ([◦ϕ]Γ , 1)). Then, it is + e ∗ easy to see that vΓ∗ is a valuation over the non-deterministic matrix (MCPL ) such that Γ ∗ vΓ (ϕ) ∈ D CPL+e iff Γ `CPL+e ϕ, for every ϕ. See the proof below: B(A

∗ Γ)

For every ϕ, ψ ∈ F or(Σ), we have:


99

• vΓ∗ (ϕ # ψ) = (([ϕ # ψ]Γ , 1), ([¬(ϕ # ψ)]Γ , 1), ([◦(ϕ # ψ)]Γ , 1)) and vΓ∗ (ϕ) # vΓ∗ (ψ) = (([ϕ]Γ , 1), ([¬ϕ]Γ , 1), ([◦ϕ]Γ , 1)) # (([ψ]Γ , 1), ([¬ψ]Γ , 1), ([◦ψ]Γ , 1)) = {(c1 , c2 , c3 ) ∈ ((AΓ )∗ )3 : c1 = ([ϕ]Γ , 1) # ([ψ]Γ , 1)}, for # ∈ {∧, ∨, →}. Since def def ([ϕ]Γ , 1) # ([ψ]Γ , 1) = ([ϕ]Γ # [ψ]Γ , 1) and [ϕ # ψ]Γ = [ϕ]Γ # [ψ]Γ , for # ∈ {∧, ∨, →}, then vΓ∗ (ϕ # ψ) ∈ vΓ∗ (ϕ) # vΓ∗ (ψ). • vΓ∗ (¬ϕ) = (([¬ϕ]Γ , 1), ([¬¬ϕ]Γ , 1), ([◦¬ϕ]Γ , 1)) and ¬(vΓ∗ (ϕ)) = ¬(([ϕ]Γ , 1), ([¬ϕ]Γ , 1), ([◦ϕ]Γ , 1)) = {(c1 , c2 , c3 ) ∈ ((AΓ )∗ )3 : ([¬ϕ]Γ , 1)}. So vΓ∗ (¬ϕ) ∈ ¬(vΓ∗ (ϕ)).

c1 =

• vΓ∗ (◦ϕ) = (([◦ϕ]Γ , 1), ([¬ ◦ ϕ]Γ , 1), ([◦ ◦ ϕ]Γ , 1)) and ◦(vΓ∗ (ϕ)) = ◦(([ϕ]Γ , 1), ([¬ϕ]Γ , 1), ([◦ϕ]Γ , 1)) = {(c1 , c2 , c3 ) ∈ ((AΓ )∗ )3 : ([◦ϕ]Γ , 1)}. So vΓ∗ (◦ϕ) ∈ ◦(vΓ∗ (ϕ)).

c1 =

+

e ∗ Thus vΓ∗ is a valuation over (MCPL ). Γ

Now, suppose that vΓ∗ (ϕ) = (([ϕ]Γ , 1), ([¬ϕ]Γ , 1), ([◦ϕ]Γ , 1)) ∈ D

CPL+ e ∗ Γ)

B(A

. Then,

([ϕ]Γ , 1) = (1Γ )∗ = (1Γ , 1) and 1Γ = [p1 ∨ ¬p1 ]Γ . So, by a similar proof to that for the Fact included in the proof of Theorem 4.1.8, we have Γ `CPL+e ϕ. If we suppose that Γ `CPL+e ϕ, again by a similar proof for the Fact mentioned above, we have [ϕ]Γ = [p1 ∨¬p1 ]Γ = 1Γ and therefore, vΓ∗ (ϕ) = (([ϕ]Γ , 1), ([¬ϕ]Γ , 1), ([◦ϕ]Γ , 1)) ∈ D CPL+e . B(A

∗ Γ)

Hence, vΓ∗ (γ) ∈ D Γ 6|=M at(KCPL+ ) α.

CPL+ e ∗ Γ)

B(A

for every γ ∈ Γ, but vΓ∗ (α) 6∈ D

CPL+ e ∗ Γ)

B(A

. From this,

e

The full subcategory in MAlg(Σ) of swap structures for CPL+ e will be denoted by SWCPL+e . That is, the class of objects of SWCPL+e is KCPL+e , and the morphisms between two given swap structures are just the homomorphisms between them as multialgebras.

4.2 Swap structures for mbC A special subclass of KCPL+e is formed by the swap structures for mbC, defined as follows: Definition 4.2.1. The universe of swap structures for mbC over a Boolean algebra A with domain A is the set BmbC = {(c1 , c2 , c3 ) ∈ A3 : c1 ∨ c2 = 1 and c1 ∧ c2 ∧ c3 = 0}. A Definition 4.2.2. Let A be a Boolean algebra. A swap structure for CPL+ e over A is said to be a swap structure for mbC over A if its domain is included in BmbC . A


100

Let KmbC = {B ∈ KCPL+e : B is a swap structure for mbC} be the class of swap structures for mbC. The following is immediate: Proposition 4.2.3. KmbC = {B ∈ KCPL+e :

|=M(B) (Ax10) ∧ (bc1)}.

Proof. Suppose that KmbC * {B ∈ KCPL+e : |=M(B) (Ax10) ∧ (bc1)} and let B be the domain of B. Then there exist a swap structure B ∈ KmbC and a valuation v over M(B) such that v((Ax10) ∧ (bc1)) 6∈ DB , that is π1 (v((Ax10) ∧ (bc1))) 6= 1. But, by definition of BmbC = {(c1 , c2 , c3 ) ∈ A3 : c1 ∨ c2 = 1 and c1 ∧ c2 ∧ c3 = 0} we have that A π1 (v((Ax10) ∧ (bc1))) = 1, see below: Since v((Ax10) ∧ (bc1)) ∈ v(Ax10) ∧ v(bc1), then π1 (v((Ax10) ∧ (bc1))) = π1 (v(Ax10))∧π1 (v(bc1)). But, π1 (v(bc1)) = π1 (v(◦α → (α → (¬α → β)))) = π1 (v(◦α)) → (π1 (v(α)) → (π1 (v(¬α)) → π1 (v(β)))) = ∼(π1 (v(◦α))∧(π1 (v(α))∧(π1 (v(¬α)))))∨π1 (v(β)). However, v(◦α) ∈ ◦v(α) and v(¬α) ∈ ¬v(α), then π1 (v(◦α)) = π3 (v(α)) and π1 (v(¬α)) = π2 (v(α)). So, ∼(π1 (v(◦α))∧(π1 (v(α))∧(π1 (v(¬α)))))∨π1 (v(β)) = ∼(π3 (v(α))∧π1 (v(α))∧ π2 (v(α))) ∨ π1 (v(β)) = ∼0 ∨ π1 (v(β)) = 1 ∨ π1 (v(β)) = 1. Similarly we show that π1 (v(Ax10)) = 1. Conversely, let B ∈ KCPL+e such that |=M(B) (Ax10) ∧ (bc1) and let p, q two different propositional variables. Let c = (c1 , c2 , c3 ) ∈ B and d ∈ π1 [B]. Consider a valuation v over M(B) such that v(p) = c and π1 (v(q)) = d. Then π1 (v(p ∨ ¬p)) = 1 and π1 (v(◦p → (p → (¬p → q)))) = 1. That is, c1 ∨ c2 = 1 and (c3 → (c1 → (c2 → d))) = 1. From this, (c1 ∧ c2 ∧ c3 ) ≤ d, for every d ∈ π1 [B]. Since by Proposition 4.1.4, π1 [B] is a classical implicative lattice included in A as subalgebra, then c1 ∧ c2 ∧ c3 = 0. From this B ∈ KmbC . mbC Given a Boolean algebra A, let BA be the unique swap structure for mbC mbC with domain BA such that, for every (a1 , a2 , a3 ) and (b1 , b2 , b3 ) in BmbC : A

(i) (a1 , a2 , a3 )#(b1 , b2 , b3 ) = {(c1 , c2 , c3 ) ∈ BmbC : c1 = a1 #b1 }, for # ∈ {∧, ∨, →}; A (ii) ¬(a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ BmbC : c1 = a2 }; A (iii) ◦(a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ BmbC : c1 = a3 }. A The full subcategory in SWCPL+e of swap structures for mbC will be denoted by SWmbC . Clearly, SWmbC is a full subcategory in MAlg(Σ). Thus, the class of objects of SWmbC is KmbC , and the morphisms between two given swap structures for mbC are the homomorphisms between them, seeing as multialgebras over Σ.


101

Let BAlg be the category of Boolean algebras defined over signature ΣBA , with Boolean algebras homomorphisms as their morphisms. Proposition 4.2.4. Let {Ai }i∈I be a family of Boolean algebras in BAlg such that, for Q every i ∈ I, Ai = hAi , ∧i , ∨i , →i , 0i , 1i i. Let A = i∈I Ai be the standard construction of the cartesian product of the family of sets {Ai }i∈I with canonical projections πi : A → Ai for every i ∈ I.3 Let A = hA, ∧, ∨, →, 0, 1i be an algebra such that its operations are given by: (i) (a#b)(i) = a(i)#i b(i), for every a, b ∈ A and # ∈ {∧, ∨, →}; (ii) 0A (i) = 0i ; (iii) 1A (i) = 1i . Then: (a) A = hA, ∧, ∨, →, 0, 1i is a Boolean algebra; (b) the canonical projections πi : A → Ai are homomorphisms of Boolean algebras; (c) hA, {πi }i∈I i is the product of the family {Ai }i∈I in BAlg. Proof. It is a well-known result that the family of Boolean algebras {Ai }i∈I has product in BAlg and that hA, {πi }i∈I i as described above is its product. Notation 4.2.5. The Boolean algebra A will be denoted by

Q

i∈I

Ai .

Consider again a family F = {Ai }i∈I of Boolean algebras such that I 6= ∅, Q and let A = i∈I Ai be its product in BAlg as described above. We want to show Q mbC that the product B = i∈I BA in MAlg(Σ) (recall Proposition 2.5.8) of the family i mbC of multialgebras {BAi }i∈I is isomorphic in MAlg(Σ) (recall Proposition 2.5.3) to the mbC multialgebra BA . To begin with, some notation is required: For i ∈ I and 1 ≤ j ≤ 3 let πji : Q (Ai )3 → Ai be the canonical projections. Observe that, if a ∈ |B| = i∈I BmbC and i ∈ I Ai Q mbC 3 then a(i) ∈ BAi ⊆ (Ai ) . Thus, for every 1 ≤ j ≤ 3 let zj ∈ i∈I Ai such that, for every i ∈ I, zj (i) = πji (a(i)). Then z = (z1 , z2 , z3 ) belongs to |A|3 . Moreover, it can be proven that z belongs to BmbC . Indeed, for every i ∈ I, z1 (i) ∨i z2 (i) = π1i (a(i)) ∨i π2i (a(i)) = 1i since A a(i) ∈ BmbC Ai . From this, z1 ∨ z2 = 1A . Analogously it can be proven that z1 ∧ z2 ∧ z3 = 0A . This allows to define a mapping fF : a∈ 3

Q

i∈I

Q

i∈I

Q BmbC → BmbC such that, for every Ai Ai i∈I

BmbC Ai , fF (a) = z where z = (z1 , z2 , z3 ) is defined as above.

I S That is, A = a ∈ : a(i) ∈ Ai for every i ∈ I , if I 6= ∅; And A is a singleton otherwise. i∈I Ai In the Chapter 0 the reader can found a definition of the cartesian product (of the family of sets).


102

Proposition 4.2.6. Let F = {Ai }i∈I be a family of Boolean algebras such that I 6= ∅. Q Q Then, the mapping fF : i∈I BmbC → BmbC is an isomorphism in MAlg(Σ). Ai Ai i∈I

Proof. Clearly fF is a bijective mapping such that its inverse mapping is given by fF−1 : Q Q BmbC → i∈I BmbC where fF−1 (z1 , z2 , z3 ) = a, with a(i) = (z1 (i), z2 (i), z3 (i)) for every Ai Ai i∈I

i ∈ I. It is also clear that, for every a, b ∈

Q

i∈I

BmbC and # ∈ {∧, ∨, →}: Ai

(i) fF [a#b] = fF (a)#fF (b); (ii) fF [¬a] = ¬fF (a); and (iii) fF [◦a] = ◦fF (a) Thus, the result follows from Proposition 2.5.3. Proposition 4.2.7. The category SWmbC has arbitrary products. Proof. Let F = {Bi }i∈I be a family of swap structures for mbC, and assume that I 6= ∅ (the case I = ∅ is trivial). By definition of KmbC , for each i ∈ I there is a Boolean algebra Ai such mbC that Bi ⊆ BA . Since SWmbC is a subcategory of MAlg(Σ) (where Σ is the signature of i mbC), and the latter has arbitrary products (cf. Proposition 2.5.8), there exists the product hB, {πi }i∈I i of F in MAlg(Σ). By the proof of Proposition 2.5.8, it is possible to define B in Q Q mbC mbC such a way that B ⊆ i∈I BA , where the multialgebra is also constructed as i∈I BAi i Q mbC in the proof of Proposition 2.5.8. Let h : B → i∈I BA be the inclusion homomorphism. i Q mbC mbC Now, let G = {Ai }i∈I and let fG : i∈I BAi → BQ Ai be the isomorphism in MAlg(Σ) i∈I

mbC of Proposition 4.2.6. Then, the homomorphism fG ◦h : B → BQ is an injective function Ai i∈I

B t

/

h

mbC BA i

Q

fG ◦h

i∈I

'

fG

mbC BQ Ai i∈I

and so it induces an isomorphism fG ◦ h in MAlg(Σ) between B and the submultialgebra mbC B 0 = (fG ◦ h)(B) of BQ , by Proposition 2.5.9. This means that hB 0 , {πi ◦ (fG ◦ h)−1 }i∈I i Ai i∈I is another realization of the product of F in MAlg(Σ). B g

fG ◦h

/

mbC BQ Ai

O

i∈I

πi

Bi o

(fG ◦h)−1 πi ◦(fG ◦h)−1

?

B0

Given that SWmbC is a full subcategory of MAlg(Σ) and by observing that B 0 is an object of SWmbC , it follows that hB 0 , {πi ◦ (fG ◦ h)−1 }i∈I i is a construction for the product in SWmbC of the family F.


103

mbC The assignment A ∈ BAlg 7→ BA ∈ SWmbC is functorial, as it will be stated in Corollary 4.2.9 below.

Proposition 4.2.8. Let f : A → A0 be a homomorphism between Boolean algebras. Then mbC mbC it induces a homomorphism f∗ : BA → BA of multialgebras given by f∗ (z1 , z2 , z3 ) = 0 (f (z1 ), f (z2 ), f (z3 )). Moreover, (f ◦ g)∗ = f∗ ◦ g∗ and (idA )∗ = idBAmbC , where idA : A → A mbC mbC and idBAmbC : BA → BA are the corresponding identity homomorphisms. Proof. Given a homomorphism f : A → A0 between Boolean algebras, let f∗ : BmbC → A mbC BA0 be the mapping such that f∗ (z) = (f (z1 ), f (z2 ), f (z3 )) for every z = (z1 , z2 , z3 ) ∈ . If z = (z1 , z2 , z3 ), w = (w1 , w2 , w3 ) ∈ BmbC and # ∈ {∧, ∨, →} then, for every BmbC A A u = (u1 , u2 , u3 ) ∈ (z#w), u1 = z1 #w1 and so f (u1 ) = f (z1 )#f (w1 ). That is, (f∗ (u))1 = (f∗ (z))1 #(f∗ (w))1 . This means that f∗ [z#w] = {f∗ (u) : u ∈ (z#w)} ⊆ {u0 ∈ BmbC : A0 0 u1 = (f∗ (z))1 #(f∗ (w))1 } = f∗ (z)#f∗ (w). On the other hand, if z = (z1 , z2 , z3 ) ∈ BmbC and u = (u1 , u2 , u3 ) ∈ ¬z then A u1 = z2 whence (f∗ (u))1 = f (u1 ) = f (z2 ) = (f∗ (z))2 . This means that f∗ (u) ∈ {u0 ∈ BmbC : u01 = (f∗ (z))2 } = ¬f∗ (z) and so f∗ [¬z] ⊆ ¬f∗ (z). Analogously it can be proven A0 mbC mbC that f∗ [◦z] ⊆ ◦f∗ (z). This shows that f∗ is indeed a homomorphism f∗ : BA → BA 0 in SWmbC . The rest of the proof is immediate, by the very definition of f∗ . mbC Corollary 4.2.9. There is a functor F : BAlg → SWmbC given by F (A) = BA for 0 every Bolean algebra A, and F (f ) = f∗ for every homomorphism f : A → A in BAlg.

Proposition 4.2.10. The functor F : BAlg → SWmbC preserves arbitrary products. Proof. It is an immediate consequence of Proposition 4.2.6 and the fact that SWmbC is a full subcategory of MAlg(Σ). Proposition 4.2.11. The functor F : BAlg → SWmbC preserves subalgebras in the mbC mbC following sense: if A is a subalgebra of A0 in BAlg then BA ⊆ BA according to 0 Definition 2.2.1. Proof. Let A and A0 be two Boolean algebras in BAlg such that A is subalgebra of A0 and let f : A → A0 be the inclusion morphism in BAlg given by f (a) = a for every a ∈ |A|. mbC mbC By Proposition 4.2.8, f induces a homomorphism f∗ : BA → BA of multialgebras 0 mbC given by f∗ (z1 , z2 , z3 ) = (f (z1 ), f (z2 ), f (z3 )) = (z1 , z2 , z3 ). By definition of |BA | and mbC 0 mbC mbC mbC mbC |BA0 | and since |A| ⊆ |A | we have |BA | ⊆ |BA0 |. Moreover, BA ⊆ BA0 .

Moreover, the following holds:


104

Proposition 4.2.12. The functor F : BAlg → SWmbC preserves monomorphisms. Proof. Let f : A → A0 be a monomorphism between Boolean algebras, and let f∗ : mbC mbC BA → BA be the induced homomorphism of multialgebras given by f∗ (z1 , z2 , z3 ) = 0 (f (z1 ), f (z2 ), f (z3 )). It is well-known that every monomorphism in BAlg is an injective function, and then f is injective. From this it is immediate to see that f∗ is also an injective function. As a consequence of Proposition 2.5.4, f∗ is a monomorphism in the category MAlg(Σ). Given that SWmbC is a full subcategory of MAlg(Σ), it follows that f∗ is a monomorphism in the category SWmbC .

As it was done in Definition 4.1.5, each B ∈ KmbC induces naturally a nondeterministic matrix M(B) = (B, DB ). Moreover, in (CARNIELLI; CONIGLIO, 2016) it was proven that the class M at(KmbC ) = {M(B) : B ∈ KmbC } semantically characterizes mbC: Theorem 4.2.13. (CARNIELLI; CONIGLIO, 2016, Theorem 6.4.8) Let Γ∪{α} ⊆ F or(Σ) be a set of formulas of mbC. Then: Γ `mbC α iff Γ |=M at(KmbC ) α. mbC The non-deterministic matrix MmbC induced by the swap structure BA 5 2 defined over the two-element Boolean algebra A2 was originally introduced by A. Avron in (AVRON, 2005)4 in order to semantically characterize the logic mbC . The domain of the n o mbC mbC multialgebra BA is the set B = t, I, t , f, f I I such that t = (1, 0, 1), I = (1, 1, 0), A2 2 tI = (1, 0, 0), f = (0, 1, 1), and fI = (0, 1, 0). Let D be the set of designated elements of the n o mbC non-deterministic matrix MmbC = M B . Then, D = {t, I, t }. Let ND = f, f I I 5 A2 be the set of non-designated truth-values. The multioperations proposed by Avron over the mbC set BmbC (see the Example 2.1.6) corresponds exactly with that for BA described after A2 2 : Proposition 4.2.3. It was proved in (AVRON, 2005) that mbC is adequate for MmbC 5

Theorem 4.2.14. (AVRON, 2005, Theorem 3.6) For every set of formulas Γ ∪ {α} ⊆ F or(Σ): Γ `mbC α iff Γ |=MmbC α. 5 As observed in (CARNIELLI; CONIGLIO, 2016, Chapter 6), Avron’s result mbC means that the non-deterministic matrix induced by the swap structure BA defined 2 over the two-element Boolean algebra A2 is sufficient for characterizing the logic mbC, and so it represents, in a certain way, the whole class KmbC of swap structures for mbC. mbC One interesting question is to prove that the 5-element multialgebra BA generates (in 2 some sense to be determined) the class KmbC , in analogy to the fact that the 2-element Boolean algebra A2 generates the class of Boolean algebras. 4

In (AVRON, 2005) Avron denoted the system mbC by B, so the non-deterministic matrix for mbC is denoted by MB 5 .


105

Recall that the power set ℘(I) of a given set I is a Boolean algebra, where the operations are the usual set-theoretic ones. A field of sets is any subalgebra of a power set Boolean algebra ℘(I). Birkhoff proves in 1935 (see (BIRKHOFF, 1935)) that every Boolean algebra A is isomorphic to a field of sets. Taking into account that ℘(I) is Q isomorphic, as a Boolean algebra, to the product i∈I A2 of Boolean algebras, Birkhoff’s result is equivalent to the following: for every Boolean algebra A, there exists a set I and Q a monomorphism of Boolean algebras h : A → i∈I A2 . From Birkhoff’s representation theorem for Boolean algebras, and taking into account the properties of the functor F : BAlg → SWmbC , a representation theorem for the class KmbC of swap structures for mbC can be obtained: Theorem 4.2.15 (Representation Theorem for KmbC ). Let B be a swap structure for ˆ : B → Qi∈I B mbC . mbC. Then, there exists a set I and a monomorphism of multialgebras h A2 Proof. Let B be a swap structure for mbC. Then, there is a Boolean algebra A such that mbC mbC B ⊆ BA . Let g : B → BA be the inclusion monomorphism in SWmbC . Using Birkhoff’s representation theorem for Boolean algebras5 , there exists a set I and a monomorphism Q h : A → i∈I A0i of Boolean algebras, where A0i = A2 , for every i ∈ I. By Proposition 4.2.12, Q mbC mbC mbC mbC → BQ there is a monomorphism h∗ : BA → BQ be the 0 . Let fG : i∈I BA0i A A0i i i∈I i∈I 0 isomorphism in MAlg(Σ) of Proposition 4.2.6, where G = {Ai }i∈I . By definition of A0i it mbC mbC ˆ : B → Qi∈I B mbC is a monomorphism follows that BA = BA , for every i ∈ I. Then h 0 A2 2 i ˆ = f −1 ◦ h∗ ◦ g. in MAlg(Σ), where h G From the previous result, it is natural to ask about the possibility of the class KmbC being a variety of multialgebras, that is, a class closed under products, submultialgebras and homomorphic images. We known that KmbC is closed under products (by Proposition 4.2.7) and submultialgebras (by the very definitions). Unfortunately, the class is not closed under homomorphic images: Proposition 4.2.16. The class KmbC of multialgebras is closed under submultialgebras and (direct) products, but it is not closed under homomorphic images. Proof. Recall the notions of multicongruence (Definition 2.4.1), quotient multialgebra (Definition 2.4.5) and the canonical map p : A → A/Θ for every multicongruence Θ n o (Proposition 2.4.8). Now, let D = {z 1 , z 2 , z 3 } and ND = z 4 , z 5 be an enumeration of mbC the elements of the domain BmbC = D ∪ ND of the multialgebra BA . Let Θ be the A2 2 equivalence relation asociated to the partition {a, b} of BmbC such that a = {z 1 , z 4 } and A2 b = {z 2 , z 3 , z 5 }. The relation Θ has the following property: for every z ∈ D there exists some w ∈ ND such that (z, w) ∈ Θ, and vice versa. From this, and by observing the definition of 5

In (DUNN; HARDEGREE, 2001, Theorem 8.11.7, p. 307).


106

mbC the multioperations in the multialgebra BA , it follows that Θ is a multicongruence over 2 mbC mbC BA2 . It is easy to prove that the multioperations in the quotient multialgebra BA /Θ 2 are trivial, that is: for every x, y ∈ {a, b} and # ∈ {∧, ∨, →}, (x#y) = ¬x = ◦x = {a, b}. mbC Clearly BA /Θ is not a swap structure for mbC: otherwise, it would generate a trivial 2 non-deterministic matrix where the set of designated values is the whole domain. This would contradict (CARNIELLI; CONIGLIO, 2016, Proposition 6.4.5(ii)), where it was proven that no non-deterministic matrix in the class M at(KmbC ) is trivial. This shows mbC mbC mbC that BA /Θ , the homomorphic image of the canonical map p : BA → BA /Θ , does 2 2 2 mbC not belong to the class KmbC , despite its domain BA is in KmbC . 2

From this last result, a question that arises is: Why don’t we change/adapt the homomorphic images or the congruence definitions in order to show that KmbC is a variety of multialgebras? The problem is that, if we do this, we will lose some important results such as the method of completeness that we apply in the main theorems of this Thesis.

4.3 Swap structures for some extensions of mbC In (CARNIELLI; CONIGLIO, 2016, Chapter 6) the concept of swap structure for mbC was generalized to some axiomatic extensions of mbC. As it was done in the Section 4.2, these structures will be reintroduced here in a slightly modified form, more suitable for an algebraic study of them. Definition 4.3.1. (CARNIELLI; CONIGLIO, 2016, Definition 3.1.1) The logic mbCciw is obtained from mbC by adding the axiom schema ◦α ∨ (α ∧ ¬α)

(ciw)

Definition 4.3.2. Let A be a Boolean algebra. The universe of swap structures for mbC mbCciw over A is the set Bciw : c3 ∨ (c1 ∧ c2 ) = 1}. A = {(c1 , c2 , c3 ) ∈ BA 3 Clearly, Bciw : c1 ∨ c2 = 1 and c3 = ∼(c1 ∧ c2 )}.6 A = {(c1 , c2 , c3 ) ∈ A

Definition 4.3.3. Let A be a Boolean algebra. A swap structure for mbC over A is said to be a swap structure for mbCciw over A if its domain is included in Bciw A . Let KmbCciw = {B ∈ KmbC : B is a swap structure for mbCciw} be the class of swap structures for mbCciw. So: Proposition 4.3.4. The following holds: KmbCciw = {B ∈ KmbC : = {B ∈ KCPL+e : 6

|=M(B) (ciw)} |=M(B) (Ax10) ∧ (bc1) ∧ (ciw)}.

Remember that, in this chapter, ∼ denotes the Boolean complement in a Bolean algebra.


107

Proof. The proof is very similar to that of Proposition 4.2.3. mbCciw For every Boolean algebra A there is a unique swap structure BA for ciw ciw mbCciw with domain BA such that, for every a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) in BA :

(i) (a1 , a2 , a3 )#(b1 , b2 , b3 ) = {(c1 , c2 , c3 ) ∈ Bciw A : c1 = a1 #b1 }, for # ∈ {∧, ∨, →}; (ii) ¬(a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ Bciw A : c1 = a2 }; (iii) ◦(a1 , a2 , a3 ) = {(c1 , c2 , c3 ) ∈ Bciw A : c1 = a3 }. The full subcategory in SWCPL+e of swap structures for mbCciw will be denoted by SWmbCciw . By the very definitions, SWmbCciw is a full subcategory in SWmbC , and a full subcategory in MAlg(Σ). Hence, the class of objects of SWmbCciw is KmbCciw , and the morphisms between two given swap structures for mbCciw are just the homomorphisms between them as multialgebras over Σ. The class M at(KmbCciw ) of non-deterministic matrices associated to swap structures for mbCciw is defined analogously to the class M at(KCPL+e ) introduced in Definition 4.1.5. Theorem 4.3.5. (CARNIELLI; CONIGLIO, 2016, Theorem 6.5.4) Let Γ ∪ {α} ⊆ F or(Σ) be a set of formulas. Then: Γ `mbCciw α iff Γ |=M at(KmbCciw ) α. Now, stronger extensions of mbC will be analized: Definition 4.3.6. Consider the following extensions of mbC: (1) The logic mbCci (CARNIELLI; CONIGLIO, 2016, Definition 3.1.7) is obtained from mbC by adding the axiom schema: ¬◦α → (α ∧ ¬α)

(ci)

(2) The logic CPLe is obtained from mbC by adding the axiom schema: ◦α

(cons)

Proposition 4.3.7. The following holds: (1) The logic mbCci properly extends mbCciw. (2) The logic CPLe is an expansion of CPL by a connective ◦ such that ◦α is a valid schema. Thus, it properly extends mbCci, and it is semantically characterized by the usual 2-valued truth-tables for CPL plus the operator ◦(x) = 1 for every x ∈ {0, 1}.


108

Proof. (1) See (CARNIELLI; CONIGLIO, 2016, Proposition 3.1.10). (2) Observe that, by (cons), (bc1) and MP, the negation ¬ is explosive in CPLe and so it coincides with the classical negation. Since CPL+ is included in CPLe then, by axiom (Ax10), this logic is nothing more than an expansion of CPL by adding as theorems all the formulas of the form ◦α. Definition 4.3.8. A swap structure for mbCci is any B ∈ KmbCciw such that: def

◦(a1 , a2 , a3 ) = {(∼(a1 ∧ a2 ), a1 ∧ a2 , 1)}. The class of swap structures for mbCci will be denoted by KmbCci . The class M at(KmbCci ) of non-deterministic matrices is defined analogously to the class M at(KCPL+e ) introduced in Definition 4.1.5. Theorem 4.3.9. (CARNIELLI; CONIGLIO, 2016, Theorem 6.5.11) Let Γ∪{α} ⊆ F or(Σ) be a set of formulas. Then: Γ `mbCci α iff Γ |=M at(KmbCci ) α. Proposition 4.3.10. The following holds: KmbCci = {B ∈ KmbCciw : = {B ∈ KmbC :

|=M(B) (ci)}

|=M(B) (ci)}

= {B ∈ KCPL+e :

|=M(B) (Ax10) ∧ (bc1) ∧ (ci)}.

Proof. The proof is very similar to that of Proposition 4.2.3. Definition 4.3.11. Let A be a Boolean algebra with domain A. The universe of swap structures for CPLe over A is the set: e BCPL = {(c1 , c2 , c3 ) ∈ Bciw : c2 = ∼c1 } = {(a, ∼a, 1) : a ∈ A} ' A. A A e Definition 4.3.12. A swap structure for CPLe is any B ∈ KmbCci such that |B| ⊆ BCPL . A The class of swap structures for CPLe will be denoted by KCPLe .

The class M at(KCPLe ) of non-deterministic matrices is defined analogously to the class M at(KCPL+e ) introduced in Definition 4.1.5. Proposition 4.3.13. The following holds: KCPLe = {B ∈ KmbCci : = {B ∈ KmbC : = {B ∈ KCPL+e :

|=M(B) (cons)} |=M(B) (cons)} |=M(B) (Ax10) ∧ (bc1) ∧ (cons)}.

Proof. The proof is very similar to that of Proposition 4.2.3.


109

CPLe mbCci For every Boolean algebra A the swap structure BA for mbCci and BA for CPLe are defined as expected. The full subcategory in SWCPL+e of swap structures for mbCci and for CPLe will be denoted by SWmbCci and SWCPLe , respectively. By the very definitions, they are full subcategories in SWmbC , and full subcategories in MAlg(Σ).

Remark 4.3.14. (1) If B ∈ KCPLe then B can be seen as a Boolean algebra isomorphic to the Boolean algebra π1 [|B|]. (2) Observe that KCPLe ⊂ KmbCci ⊂ KmbCciw ⊂ KmbC ⊂ KCPL+e while CPLe ⊃ mbCci ⊃ mbCciw ⊃ mbC ⊃ CPL+ e. As analyzed in (CARNIELLI; CONIGLIO, 2016, Chapter 6), the logic mbCciw can be characterized by a single 3-valued non-deterministic matrix, by considering the twoelement Boolean algebra A2 . Indeed the non-deterministic matrix MmbCciw induced by the 3 mbCciw mbCciw mbCciw swap structure BA2 , that is M3 = M BA2 , was originally considered by A. Avron in (AVRON, 2005) obtaining so a semantical characterization of mbCciw. The n o mbCciw mbCciw domain of the multialgebra BA is the set B = t, I, f such that t = (1, 0, 1), A2 2 I = (1, 1, 0) and f = (0, 1, 1), where D3 = {t, I} is the set of designated elements of the non-deterministic matrix MmbCciw . The multioperations are defined as follows: 3 ∧

t

I

f

∨

t

I

f

t

{t, I}

{t, I}

{F }

t

{t, I}

{t, I}

{t, I}

I

{t, I}

{t, I}

{F }

I

{t, I}

{t, I}

{t, I}

f

{F }

{F }

{F }

f

{t, I}

{t, I}

{F }

→

t

I

f

¬

◦

t

{t, I}

{t, I}

{f }

t

{f }

t

{t, I}

I

{t, I}

{t, I}

{f }

I

{t, I}

I

{f }

f

{t, I}

{t, I}

{t, I}

f

{t, I}

f

{t, I}

mbCciw mbC It is clear that BA is a submultialgebra of BA . Moreover, by an analysis 2 2 similar to the one presented above, it is possible to prove representation theorem for KmbCciw analogous to that for KmbC (recall Theorem 4.2.15). In order to do this, consider the following lemma:


110

Lemma 4.3.15. Let an assignment FmbCciw : BAlg → SWmbCciw , with FmbCciw (A) = mbCciw BA , and FmbCciw (f ) = f∗ for every morphism f : A → A0 in BAlg, where f∗ : mbCciw mbCciw BA → BA and f∗ (z) = (f (z1 ), f (z2 ), f (z3 )), for every z ∈ BmbCciw . Then 0 A FmbCciw is a functor which preserves monomorphisms and arbitrary products. mbCciw mbC Proof. From BA ⊆ BA (as submultialgebra) and by proof of the Proposition 4.2.8 (in the case of F : BAlg → SWmbC ), it is easy to see that f∗ is a morphism in SWmbCciw . So, FmbCciw is a functor.

By Proposition 4.2.6 and the fact that SWmbCciw is a full subcategory of SWmbC that is a full subcategory of MAlg(Σ), then the functor FmbCciw : BAlg → SWmbCciw preserves arbitrary products. Let f : A → A0 be a monomorphism in BAlg. It is well-known that every monomorphism in BAlg is an injective function, and then f is injective. From this it is immediate to see that f∗ is also an injective function. As a consequence of Proposition 2.5.4, f∗ is a monomorphism in the category MAlg(Σ). Given that SWmbCciw is a full subcategory of SWmbC that is a full subcategory of MAlg(Σ), it follows that f∗ is a monomorphism in SWmbCciw . So, the functor FmbCciw : BAlg → SWmbCciw preserves monomorphisms. Theorem 4.3.16 (Representation Theorem for KmbCciw ). Let B be a swap structure ˆ:B→ for mbCciw. Then, there exists a set I and a monomorphism of multialgebras h Q mbCciw . i∈I BA2 Proof. Let B be a swap structure for mbCciw. Then, there is a Boolean algebra A such mbCciw mbCciw that B ⊆ BA . Let g : B → BA be the inclusion monomorphism in SWmbCciw . Using Birkhoff’s representation theorem for Boolean algebras7 , there exists a set I and a Q monomorphism h : A → i∈I A0i of Boolean algebras, where A0i = A2 , for every i ∈ I. Consider the functor FmbCciw : BAlg → SWmbCciw such that FmbCciw (h) = h∗ which preserves monomorphisms (by Lemma 4.3.15). So, there is a monomorphism mbCciw mbCciw h∗ : BA → BQ . A0 i∈I

i

Recall that SWmbCciw is a full subcategory in SWmbC and SWmbC is a Q mbCciw mbCciw full subcategory in MAlg(Σ), and we have that fG : i∈I BA → BQ is an 0 A0i i i∈I 0 isomorphism in MAlg(Σ), where G = {Ai }i∈I (by Proposition 4.2.6). By definition of mbCciw mbCciw ˆ : B → Qi∈I B mbCciw is a A0i it follows that BA = BA , for every i ∈ I. Then h 0 A2 2 i −1 ˆ monomorphism in MAlg(Σ), where h = fG ◦ h∗ ◦ g. 7

In (DUNN; HARDEGREE, 2001, Theorem 8.11.7, p. 307).


111

Concerning mbCci and CPLe , similar results can be obtained. Indeed, A. Avron has proven in (AVRON, 2005) that mbCci can be characterized by a single 3-valued non-deterministic matrix MmbCci . 3 In (CARNIELLI; CONIGLIO, 2016, Chapter 6) was proved that MmbCci is 3 mbCci mbCci mbCci the one obtained by the 3-valued swap structure BA2 , that is M3 = M BA2 , mbCciw with the same domain and multioperations than BA2 , but now the multioperator ◦ is single-valued, and it is defined as follows: ◦ t

{t}

I

{f }

f

{t}

mbCci mbCciw mbC Clearly, BA is a submultialgebra of BA and so of BA . Moreover: 2 2 2

Theorem 4.3.17 (Representation Theorem for KmbCci ). Let B be a swap structure for ˆ : B → mbCci. Then, there exists a set I and a monomorphism of multialgebras h Q mbCci . i∈I BA2 Proof. It is enough to observe that, let f : A → A0 be a monomorphism in BAlg and let FmbCci : BAlg → SWmbCci such that FmbCci (f ) = f∗ be a functor that preserves monomorphisms (by similar proof of the Lemma 4.3.15). If a ∈ BmbCci , A def then ◦a = {(∼(a1 ∧ a2 ), a1 ∧ a2 , 1)}. So, f∗ [◦a] = {f∗ (∼(a1 ∧ a2 ), a1 ∧ a2 , 1)} = {(f (∼(a1 ∧ a2 )), f (a1 ∧ a2 ), f (1))} = {(∼(f (a1 ) ∧ f (a2 )), (f (a1 ) ∧ f (a2 )), f (1))} = ◦f∗ (a) and therefore f∗ is a monomorphism in SWmbCci . The remainder of this proof is similar to proof of Theorem 4.3.16. CPLe Finally, the case of CPLe is quite simple. The swap structure BA has 2 domain {t, f } where t = (1, 0, 1) and f = (0, 1, 1). The multiperations are single-valued, producing a Boolean algebra isomorphic to A2 . +

CPLe CPLe mbCci mbCciw mbC Clearly BA ⊆ BA ⊆ BA ⊆ BA ⊆ BA . Additionally: 2 2 2 2 2

Theorem 4.3.18 (Representation Theorem for KCPLe ). Let B be a swap structure for ˆ : B → CPLe . Then, there exists a set I and a monomorphism of multialgebras h Q CPLe . i∈I BA2 Proof. It is similar to proof of Theorem 4.3.16.

The last theorem is just the original Birkhoff’s theorem for Boolean algebras published in 1935, under a different presentation.

112

Final Considerations This Thesis proposes a general study of non-deterministic matrices applied to logic systems from the perspective of Universal algebra and category theory. The non-deterministic matrices differ from usual matrices (deterministic) by the use of what is called today “multioperations”. That is, operations which assign, to some element of the domain, a non-empty subset of that domain. When this Thesis was conceived while a research project, we aimed to develop a formal theory for non-deterministic matrices semantics in order to better understand the scope of the original proposal by Avron and his collaborators. The main goal of the research was to propose an alternative way for algebraization of logic systems, in order to deal with logics in which the usual algebraization methods cannot (or it is very hard to) be applied. Starting from this, the concept of “non-deterministic algebra” (or “ND-algebra”)8 arose naturally. However, soon we realize that this notion was already proposed and intensively studied in the literature, under different names: “hyperalgebras”, “multialgebras” and “non-deterministic algebras”, among others. Since non-deterministic algebras constitute a generalization of standard algebras, it is natural that the generalization of the usual concepts for this framework is not unique. Being so, at least three different definitions of multialgebra were proposed, and this situation is similar for more specific concepts, such as homomorphism, for which at least five distinct definitions are available. Thus, after adapting and organizing the diverse nomenclature for these concepts, we present them in Chapter 2. Since there are many possibilities for each concept, the choice of the “right” notion in each case was motivated by purely pragmatical reasons: the notions better suited to our proposals (namely, its application to the study of logic systems) were adopted. It is important to note that most of the authors have studied and developed the theory of hyperstructures independently, which could justify the different names for similar concepts. From this point of view, the historical analysis obtained in Chapter 1 was extremely useful for us. In fact, this historical research leads us to the discovery of some interesting facts: for instance, that Marty in 1935 (MARTY, 1935) already introduced a definition of homomorphism between hyperstructures and so, probably he was the first to present this concept. We also discovered that the Brazilian logician and mathematician Antonio Antunes Mario Sette (from the Centre for Logic, Epistemology and the History of Science –CLE at the University of Campinas – UNICAMP) already used in his Master’s thesis from 1971 the concept of hyperlattice under the name of “reticuloide”. We also found 8

By analogy with the terms “non-deterministic matrix” (or ND-matrix) and “Nmatrix” coined by Avron and his collaborators (AVRON; LEV, 2001).

Final Considerations

113

that the concept of non-deterministic matrices was applied by other authors much before the works of Avron and his colaborators (AVRON; LEV, 2001). This study provided several possibilities for several future developments in the study of multialgebras applied to Logic. For instance, the concepts presented in Chapter 2 enabled the development in Chapter 4 of an incipient algebraic theory of swap structures (a particular class of multialgebras introduced by Carnielli and Coniglio in (CARNIELLI; CONIGLIO, 2016, Chapter 6)), by adapting concepts of universal algebra to multialgebras in a suitable way. Although we found in the literature a large amount of theory and research about multialgebras, its application to Logic and especially to the algebraization of logic systems was not enough explored, as far as we know. In Chapter 3 we introduced an original class of swap structures as a suitable semantics for a family of non-normal modal systems. This is the first example of swap structures defined for logics outside the scope of Logics of Formal Inconsistency (LFIs), the class of paraconsistent logics for which swap structures were originally proposed. Based on the ideas of the completeness proofs of LFIs with respect to Fidel structures semantics found in (CARNIELLI; CONIGLIO, 2016, Chapter 6), a quotient swap structure that we called of Lindenbaum-Tarski swap structure, as well as a canonical valuation over it, were proposed. After defining a suitable notion of swap structures, as well as the class of non-deterministic matrices naturally associated to them, it was possible to prove the soundness theorems of the Hilbert calculi defining these non-normal modal systems with respect to such Nmatrix semantics. In order to obtain the completeness theorem, the method of Lindenbaum-Tarski swap structures mentioned above was employed. The notion of Lindenbaum-Tarski swap structures proposed here generalizes in a quite natural way the classical Lindenbaum-Tarski method, allowing to deal with logics which are not algebraizable in the usual sense. We conjecture that this technique could be applied to a wide class of non-algebraizable logics (even by the general techniques of Blok-Pigozzi), allowing an interesting and new paradigm for algebraizing logics by means of multialgebras. The Lindenbaum-Tarski swap structures method was also applied in Chapter 4 in order to obtain completeness theorem for the system CPL+ e (see Theorems 4.1.8 and 4.1.13). Its application to all the LFIs studied there should be immediate (observe that the adequacy of such LFIs with respect to swap structures semantics was already stated in (CARNIELLI; CONIGLIO, 2016, Chapter 6), by proving its equivalence with the Fidel structures semantics). Besides this, in Chapter 4 other important results were obtained, concerning the algebraic and categorial properties of several categories of swap structures (seen as multialgebras), including representation theorems analogous to Birkhoff’s theorem for Boolean algebras.


114

Finally, some suggestions for future work that arise naturally from what has been developed in the present Thesis will be presented: • In this Thesis, the method of Lindenbaum-Tarski swap structures was applied to two distinct family of logic systems (namely, non-normal modal logics and LFIs). They have in common the existence of at least one non-congruential operator, justifying so this non-deterministic algebraic approach. We believe that the same method can be applied to other logical systems, with similar conditions (that is, where the application of the usual algebraic methods is hard or even impossible). • The behavior of the logics presented in Chapter 3 and the definitions of swap structures and of Lindenbaum-Tarski swap structures for them suggests that it is possible to obtain algebraic and categorial results, similar to those obtained in Chapter 4 for the family of LFIs, to the family of non-normal modal systems presented in Chapter 3 of this thesis. That is, a modular treatment of the algebraic theory of swap structures for these modal logics could be obtained in this way. • In the paper (SCHWEIGERT, 1985) by Schweigert the proof of the Birkhoff’s theorem for multialgebras9 is clearly incomplete. Indeed, in that paper the author does not specify the basic definitions that are being adopted in order to obtain the main result. Given that, as discussed above (and in Chapter 2 of this Thesis), each concept from ordinary algebra can be generalized in several ways to the realm of multialgebras, this is not a minor issue. By its turn, Hansoul (HANSOUL, 1983) presented a detailed proof of a version of the Birkhoff’s theorem for multialgebras. However, the definitions adopted there are too restrictive for our purpose, namely, its application to Logic. For instance, the author assumes that any multioperation must return a finite set of possible values for each argument. In view of this, it would be interesting to investigate the validity of the Birkhoff’s theorem for multialgebras using the definitions presented in Chapter 2 of this thesis. • Some results have been obtained for the category of multialgebras proposed in Chapter 2. In Chapter 1 we presented some other general proposals and, in Chapter 4, we developed theoretic studies applied to some LFIs. We can still develop an algebraic and categorial study of swap structures (and other categories of multialgebras associated to logic systems) by changing the definition of homomorphism, similar to what Nolan did in (NOLAN, 1979). As mentioned above, it is important to observe that the theory of multialgebras was not explored from the point of view of its logical application. In this sense, the present 9

Every multialgebra can be represented as a sub-direct product of sub-directly irreducible multialgebras.


115

research could be considered as pioneering, and the results presented in chapters 3 and 4 speak by themselves. Thus, this field has several open possibilities. This promises to be a fruitful ground for new researches.

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