Logic, Algebra and Truth Degrees (LATD) Working group on Mathematical Fuzzy Logic
Second conference LATD September 7-11, 2010, Prague, Czech Republic 1
Program Committee1, Petr Hájek (Chair) Antonio Di Nola Christian Fermüller Siegfried Gottwald Daniele Mundici Carles Noguera Aleš Pultr Organizing Committee Petr Cintula (Chair) Karel Chvalovský Petra Ivanicová Milan Petrík Aleš Pultr Martin Víta
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Home page: http://www.mathfuzzlog.org/latd2010/index.php
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BISYMMETRIC GÖDEL ALGEBRAS WITH SPECIAL MODAL OPERATORS
Mircea Sularia2
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Faculty of Applied Sciences, Department of Mathematics II, Polytechnic University of Bucharest, Romania, e-mail:
[email protected].
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Abstract. A Gödel-Kleene algebra (LK -algebra) is a symmetric Gödel algebra (SG-algebra) such that its De Morgan algebra reduct is a Kleene algebra. An essential fact established by Monteiro is that every LK -algebra is isomorphic to a subdirect product of SG-chains. Then a structure called bisymmetric Gödel algebra (S2G-algebra) is de ned by an LK -algebra equipped with an involutive automorphism (Moisil symmetry). A special class of S2G-algebras called modal bisymmetric Gödel algebras ( S2G-algebras) is introduced. On this basis one obtains an algebraic semantics for a sentential modal logic of bicriteria decision making. Then the development of this study for the other more interesting cases of multicriteria decision making is also considered in connection with mathematical fuzzy logic.
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CONTENTS
Preliminary de nitions The Boolean centre of symmetric Gödel algebra Subdirectly irreducible LK-algebras Bisymmetric Gödel algebras (S2Galgebras) Bidimensional Brouwerian algebras (BrD-algebras) Free BrD-algebras construction Modal S2G-algebras and concluding remarks 5
1 Preliminary de nitions De nition 1 A Heyting algebra [1; 2] is an algebra H = (H; _; ^; !; 0; 1) of type H = (2; 2; 2; 0; 0) such that (H; _; ^; 0; 1) is a lattice with zero 0 and one 1 satisfying the following condition, for all x; y; z 2 H : (H) z
x ! y if and only if z ^ x
y.
[1] Balbes, R. and Dwinger, Ph., Distributive Lattices, Columbia University, 1974. [2] Blyth, T. S., Lattices and algebraic structures, Springer, 2008. De nition 2 A Gödel algebra is a Heyting algebra G = (G; _; ^; !; 0; 1) verifying the prelinearity condition, for all x; y 2 G: (D) (x ! y) _ (y ! x) = 1: 6
Remark 1 A Gödel algebra [3; 4] is called an L-algebra in [1; 5] or a Linear Heyting algebra in [6].
[3] Gottwald, S., Mathematical fuzzy logic as a tool for the treatment of vague information, Information Sciences, Vol. 172, 2005, pp. 41-71. [4] Hájek, P., Metamathematics of Fuzzy Logic, Trends in Logic, vol. 4, Kluwer Academic Publishers, Dordrecht, 1998. [5] Horn, A., Logic with truth values in a linearly ordered Heyting algebra, J. Symbolic Logic 34 (1969) 395-408. [6] Martínez, N. G. and Priestley, H. A., On Priestley spaces of lattice-ordered algebraic structures, Order 15 (1998) 297-323. 7
De nition 3 A dual Heyting algebra is an algebra B = (B; _; ^; ; 0; 1) of type H such that (B; _; ^; 0; 1) is a bounded lattice and the following condition holds, for all x; y; z 2 B : (Ho) x y z if and only if x y _ z .
De nition 4 An algebra such that it is either a Heyting algebra or a dual Heyting algebra will be called a bounded Brouwerian algebra.
De nition 5 A double Heyting algebra is dened by a system A = (A; _; ^; !; ; 0; 1) such that H(A) = (A; _; ^; !; 0; 1) is a Heyting algebra and H o(A) = (A; _; ^; ; 0; 1) is a dual Heyting algebra. 8
De nition 6 A symmetric Heyting algebra (or an SH-algebra) is a Heyting algebra together with a De Morgan negation.
De nition 7 Any SH -algebra such that its lattice reduct is a chain will be called an SH chain. In the sequel any SH-algebra is an algebra A = (A; _; ^; !; ; 0; 1) of type SH = (2; 2; 2; 1; 0; 0) satisfying: (Hr) H (A) = (A; _; ^; !; 0; 1) is a Heyting algebra, (M ) the operator is a De Morgan negation of (A; _; ^; 0; 1) i.e. (M 1) (x ) = x; (M 2) (x _ y) = x ^ y ; (M 3) (x ^ y) = x _ y . The condition (M 1) is called the law of double negation and the conditions (M 2) and (M 3) are called the De Morgan laws. 9
2 The Boolean centre of symmetric Gödel algebra
De nition 8 A symmetric Gödel algebra (or an SG-algebra) is a Gödel algebra together with a De Morgan negation.
De nition 9 An SG-algebra such that its lattice reduct is a chain will be called an SGchain. Any SG-algebra is an SH-algebra as above G = (G; _; ^; !; ; 0; 1) such that its Heyting algebra reduct H(G ) = (G; _; ^; !; 0; 1) is a Gödel algebra (previous De nitions). Any SH-chain is an SG-chain. 10
Remark 2 A double Heyting algebra is called a Heyting-Brouwer algebra in [7] or a semiBoolean algebra in [8]. The structures of SHalgebra and SG-algebra are introduced in [9].
[7] Iturrioz, L., Les algèbres de HeytingBrouwer et de ukasiewicz trivalentes, Notre-Dame Journal of Formal Logic, Vol. XVII, No. 1, 1976, pp. 119-126. [8] Rauszer, C., An algebraic and Kripke style approach to certain extensions of intuitionistic logic, Dissertationes Mathematicae, CLXII, 1980, pp. 1-67. [9] Monteiro, A., Sur les algèbres de Heyting symétriques, Portugaliae Mathematica 39 (1-4) (1980) 1-237. 11
In accordance with the de nition of involution lattice presented in [2], an SHalgebra (SG-algebra) also can be called an involution Heyting (Gödel) algebra since it is a Heyting (Gödel) algebra such that its lattice reduct is an involution lattice.
Now we introduce the notion of central element of an SG-algebra.
De nition 10 Let G be an SG-algebra. A central element of G is any c 2 G such that there are a pair of SG-algebras (A ; B ) and an isomorphism f : G ! A B satisfying f (c) = (1; 0) or f (c) = (0; 1). 12
Notation 1 For any SG-algebra G = (G; _; ^; !; ; 0; 1), let L(G ) be the bounded lattice reduct of G , B(G ) be the set of complemented elements of L(G ) and Z(G ) be the set of central elements of G . If x; y; ; 2 G with then we let G[ ; ] = fx 2 G = x g and we will use the following abbreviations:
(1) :x = x ! 0, (2) x y = (y ! x ) , (3) :ox = 1 x.
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Now we present auxiliary results in order to describe the two sets B(G ) and Z(G ) previously de ned. Lemma 1 If G = (G; _; ^; !; ; 0; 1) is an SG-algebra and :G = f:x = x 2 Gg then the next conditions hold: (i) B(G ) = :G, (ii) HBr(G ) = (G; _; ^; !; ; 0; 1) is a double Heyting algebra. Proof. We have H(G ) = (G; _; ^; !; 0; 1) is a Gödel algebra, This implies that for all a 2 G : a 2 B(G ) iff (9b 2 G) a^b = 0 and a _ b = 1 iff (9x 2 G) a = :x iff a 2 :G. Thus (i) holds. The condition (ii) follows from previous de nitions using the relation (Ho), where for all x; y 2 G the element x y is de ned by (2) [Notation]. The relation (Ho) holds since if x; y; z 2 G then using (H) we have x y z iff (y ! x ) z iff z (y ! x ) = y ! x iff z ^y x iff x = x (z ^ y ) = y _ z . 14
Lemma 2 In any SG-algebra G the following dual prelinearity condition is satis ed, for all x; y 2 G: (Do) (x y) ^ (y x) = 0.
Proof. If x; y 2 G then using relations (D) and (2) one derives (x y) ^ (y x) = (y ! x ) ^ (x ! y ) = ((y ! x ) _ (x ! y )) = 1 = 0:
Remark 3 If G = (G; _; ^; !; ; 0; 1) is an SG-algebra and G[ ; ] ( ) is an interval then the there exists a Gödel algebra G [ ; ] = G[ ; ]; _; ^; ! ; 0 ; 1 such that 0 = , 1 = and for all x; y 2 G[ ; ] one de nes x ! y = (x ! y) ^ : 15
Lemma 3 Let G be an SG-algebra, a 2 G and the interval G[a] = G[a ^ a ; a] Then the a system G [a] = G[a]; _; ^; !a; ; 0a; 1a is an SG-algebra, where for all x; y 2 G[a], one a de nes x !a y = (x ! y) ^ a, x = x ^ a, 0a = a ^ a and 1a = a.
Proof. For all x; y 2 G[a], we have a a G[a]. The reduct fx ! y; x ; 0a; 1ag H(G [a]) = (G[a]; _; ^; !a; 0a; 1a) is just the Gödel algebra G [ ; ] from Remark 3, where = a ^ a and = a. One veri es a that the operator : G[a] ! G[a] is a De Morgan negation of the bounded lattice reduct L (G [a]) = (G[a]; _; ^; 0a; 1a). Using De nition 8 one derives that G [a] is an SG-algebra. 16
Lemma 4 For any SG-algebra G we have Z(G ) fc 2 B(G ) = c ^ c = 0g.
Proof. Let c 2 Z(G ) (previous Notation). De nition 10 implies that there are a pair of SG-algebras (A ; B ) and an isomorphism f :G !A B such that f (c) = (1; 0) or f (c) = (0; 1). Then c 2 B(G ) and c ^ c = 0 since f is an isomorphism and both the element a = (1; 0) and the element a = (0; 1) are complemented such that a ^ a = (0; 0) = a ^ a in A B. 17
Lemma 5 For any SG-algebra G we have fc 2 B(G ) = c ^ c = 0g Z(G ). Proof. Let c 2 B(G ) such that c ^ c = 0. Then the element d = c also satis es d 2 B(G ) and d^d = 0 since c^d = c^c = 0, c _ d = c _ c = (c ^ c ) = 0 = 1 and d ^ d = c ^ c = c ^ c = 0. Now we associate with the pair (c; d) a pair of SG-algebras (A ; B ), where A = G [c] and B = G [d] are de ned in Lemma 3. From c ^ c = 0 and d ^ d = 0 it follows that A = (A; _; ^; !A; A ; 0A; 1A) and B = (B; _; ^; !B ; B ; 0B ; 1B ), where A = G[c] = G[c ^ c ; c] = G[0; c], B = G[d] = G[d ^ d ; d] = G[0; d] and for all a; a0 2 A and b; b0 2 B , a !A a0 = (a ! a0) ^ c, a A = a ^ c, 0A = 0, 1A = c, b !B b0 = (b ! b0) ^ d, b B = b ^ d, 0B = 0, 1B = d. De ne a function f : G ! A B such that for all x 2 G; f (x) = (x ^ c; x ^ d). Then f (c) = (c; c ^ d) = (c; 0) = (1A; 0B ). One can verify that f is an isomorphism 18
of SG-algebras from G onto A B. Therefore using De nition 10 it follows that c is a central element of G i.e. c 2 Z(G ). Theorem 6 If G is an SG-algebra then we have Z(G ) = fc 2 B(G ) = c ^ c = 0g. Proof. This result follows from the above Lemmas. Corollary 7 For any SG-algebra G , the set of its central elements Z(G ) is a Boolean algebra Z(G ) = (Z(G ); _; ^;s ; 0; 1), where if c 2 Z(G ) then the complement of c is given by cs = c . Proof. Theorem 6 implies that for all c; d 2 Z(G ), fc _ d; c ^ d; cs; 0; 1g Z(G ), c ^ cs = 0 and c _ cs = 1. Thus Z(G ) is a sublattice of L(G ) together with the complementation operator s: 19
Corollary 8 Let G be an SG-algebra such that for all c 2 B(G ), c^c = 0. Then the set of its central elements satis es Z(G ) = :G, where :G is the support set of the Boolean algebra :G = (:G; _; ^;c ; 0; 1) such that if x 2 :G then the complement of x is given by xc = :x. Proof. Suppose (8c 2 B(G )) c ^ c = 0. Theorem 6 implies Z(G ) = B(G ), but using Lemma 1 (i) we have B(G ) = :G, thus Z(G ) = :G. The system :G is a Boolean algebra since the system St(G ) = (G; _; ^; :; 0; 1) is a Stone algebra. In addition Corollary 7 implies that in this case (8x 2 :G) xc = xs, since the complement of x 2 Z(G ) = :G is unique. De nition 11 Let G be an SG-algebra. The Boolean algebra Z(G ) (Corollary 7) will be called the Boolean centre of G . The Boolean algebra :G (Corollary 8) will be called the Boolean centre of L(G ). 20
3 Subdirectly irreducible LK-algebras The relation Z(G ) B(G ) holds, but Z(G ) 6= B(G ) for some SG-algebras (an example is the symmetric Boolean algebra with four elements G = B4 ). The starting point of this paper is the consideration of the variety of SG-algebras G called LK -algebras [9] such that the equality Z(G ) = B(G ) holds. In this section we present this variety and an essential fact on the Birkhoff subdirect representation theorem in this case. The main result is the characterization of subdirectly irreducible LK-algebras established by Monteiro in [9]. De nition 12 A Gödel-Kleene algebra (LK algebra) is an SG-algebra G such that the following relation holds, for all x; y 2 G : (K) x ^ x
y_y . 21
Remark 4 The notion of LK-algebra is introduced in [9] by an SH-algebra A such that its Heyting algebra reduct H(A) is a Gödel algebra (L-algebra) and its de Morgan algebra reduct M (G ) = (G; _; ^; ; 0; 1) is a Kleene algebra (K -algebra). Thus, if one uses the term Gödel algebra for L-algebra then it is justi ed to use in De nition 12 the term GödelKleene algebra for LK-algebra. The next result shows that the class of LK-algebras is a class of SG-algebras G such that the set of its complemented elements B(G ) is a subalgebra of its De Morgan algebra reduct M (G ).
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Theorem 9 Let G = (G; _; ^; !; ; 0; 1) be an SG-algebra. Then the following conditions are equivalent:
(i) G is an LK-algebra. (ii) :x x , for all x 2 G: (iii) (:x) = ::x, for all x 2 G. Proof. This result is presented in [9]. We also obtain the center of LK-algebra. Theorem 10 If G is an LK-algebra then we have Z(G ) = B(G ). Proof. Let G be an LK-algebra. From De nition 12 and Theorem 9 it follows that G is an SG-algebra.satisfying 9 (iii). Using Lemma 1 (i) this implies the relation B(G ) = fc 2 G = c ^ c = 0g = :G. From Corollary 8 one obtains Z(G ) = B(G ). 23
The next theorem presents an equational theory of LK-algebras. Theorem 11 Let G = (G; _; ^; !; ; 0; 1) be an algebra.of type SH . The following conditions are equivalent: (i) G is an LK-algebra. (ii) For all x; y; z 2 G:
(1) x ^ (x _ y) = x; (2) x^(y_z) = (z^(x_0))_(y^(x_0)); (3) x ^ (x ! y) = x ^ y; (4) x ! (y ! z) = (x ^ y) ! z; (5) (x ! y) _ (y ! x) = 1; (6) (x ^ y ) = x _ y; (7) (:x) = ::x:
Proof. One can verify the following properties: (p1) the reduct (G; _; ^; 0) is a distributive lattice with zero 0 iff it satis es (1) (2); (p2) G is an SG-algebra iff G satis es (1) (6); (p3) if G is an SG-algebra then :G is a subalgebra of M (G ) iff G 24
satis es (7) iff G satis es (K) (De nition 12). Property (p1) is also presented in [10; 11]. Property (p2) follows from (p1) and De nitions 2 and 8. Property (p3) follows from Theorem 9. Suppose that (i) holds. Then G is an SG-algebra satisfying the relation (7). Thus (ii) holds since the identities (1) (6) also are valid in G by (p2): Now suppose (ii) i.e. identities (1) (7) are valid in G . Then (i) follows from De nition 12 using (p2) and (p3). [10] Padmanabhan, R. and Rudeanu, S., Axioms for Lattices and Boolean Algebras, World Scienti c, New Jersey - London Singapore, 2008. [11] Sholander, M., Postulates for distributive lattices, Canadian Journal of Mathematics, 3 (1951) 28-30. The next result is a characterization of subdirectly irreducible LK-algebras. 25
Theorem 12 An LK-algebra is subdirectly irreducible iff it is an SG-chain.
Proof. This result is also presented in [9].
Corollary 13 Any LK-algebra is isomorphic to a subdirect product of SG-chains.
Proof. Theorems 11, 12 and Birkhoff representation theorem imply the corollary.
4 Bisymmetric Gödel algebras (S2G-algebras) In this section we consider a structure called bisymmetric Gödel algebra (S2G-algebra) de ned by an LK-algebra equipped with an involutive automorphism. The next de nition presents this special notion. 26
De nition 13 An S2G-algebra is a de ned by a system A = (A; _; ^; !; ; ; 0; 1), where A is a nonempty set together with three binary operations _; ^; !, two operators ; and two constants 0; 1 such that the following conditions hold: (s1) the reduct G (A) = (A; _; ^; !; ; 0; 1) is an LK-algebra, (s2) the operator : A ! A is an involutive automorphism of G (A). If A is an S2G-algebra then Con(A) is its lattice of congruences and Fn(A) is the lattice of normal lters of A i.e. F 2 Fn(A) iff F is a lter of the lattice reduct L(A) = (A; _; ^; 0; 1) such that the equivalence relation rF de ned by F is compatible with the two operators and . Remark 5 The condition (s1) holds iff G (A) satis es (1) (7) (Theorem 11 (ii)), for all x; y; z 2 A. The condition (s2) expresses the 27
fact that is a symmetry in the sense of Moisil [12-14] which is called by Monteiro a 2-cyclic automorphism. Thus an S2G-algebra is a 2cyclic LK-algebra [14].
[12] Moisil, Gr. C., Algebra of circuits with valves (Algebra schemelor cu elemente ventil), Revista Universita¸tii C. I. Parhon s¸i a Politehnicii Bucure¸sti 4-5 (1954) 9-42 (in Romanian). [13] Díaz Varela, J. P., Equivalence between varieties of square root rings and Boolean algebras with a distinguished automorphism, Journal of Algebra, 299 (2003) 190-197. [14] Abad, M., Díaz Varela, J. P., Fernández, A., Meske, N. and Rueda, L., On cyclic symmetric Heyting algebras, Portugaliae Mathematica, 58 (2001) 389-406. 28
Theorem 14 If A = (A; _; ^; !; ; ; 0; 1) then the following conditions are equivalent:
(i) A is an S2G-algebra, (ii) A satis es (1) (7) (Theorem 11 (ii)) together with the following conditions, for all x; y 2 A: (8) ( (x) ^ (y)) = x ^ y; (9) (x ! y) = (x) ! (y); (10) (x ) = ( (x)) : Corollary 15 Let A be an S2G-algebra. Then for all x; y; z 2 A:
x ! (y _ z) = (x ! y) _ (x ! z); x ! (y ^ z) = (x ! y) ^ (x ! z); (x _ y) ! z = (x ! z) ^ (y ! z); (x ^ y) ! z = (x ! z) _ (y ! z); x _ y = ((x ! y) ! y) ^ ((y ! x) ! x); ( (x)) = x; (x ^ y) = (x) ^ (y); 29
(x _ y) = (x) _ (y); (1) = 1, (0) = 0; :x ! x = 1 :(x ! y) ! (y ! x ) = 1:
The two sets of congruences Con(A) and of normal lters Fn(A) are isomorphic complete lattices.
5 Brouwerian D-algebras Let D = (2; 2; 2; 2; 0; 0) be the type of : algebras with binary operations _; ^; !; and constants 0; 1. We introduce a variety D K( D ) of algebras called Brouwerian D-algebras (BrD-algebras) [15-18]. 30
[15] Sularia, M., Sur une extension de la classe des algèbres de Heyting, C.R. Acad. Sci. Paris Ser.I, t.302, 1986, pp. 83-86. [16] Sularia, M., Involutive Brouwerian D-algebras, WSEAS Trans. Math., Issue 10, Volume 5 (2006) 1108-1116. [17] Sularia, M., Partially ordered structures including symmetric Boolean algebras, LATD First conference of the working group on Mathematical Fuzzy Logic, 8-11 September 2008, Siena, Italy. [18] Sularia, M., A minimal variety including bounded Brouwerian algebras, BLAST International Conference, August 10-14, 2009, New Mexico State University, USA [http://subsessile.nmsu.edu/blast]. An equational de nition of D is given. One derives that D is the minimal variety of algebras in K( D ) including all bounded Brouwerian algebras (De nition 4). Then other basic properties of D are presented. 31
Notation 2 We will use special notations for some variety of algebras:
C
(1) B - Boolean algebras, (2) H - Heyting algebras, (3) Br - dual Heyting algebras, (4) HBr - double Heyting algebras, (5) SH - SH-algebras, (6) SG - SG-algebras, (7) hCi is the variety generated by a class K( ), for any type :
De nition 14 If B = (B;: _; ^;c ; 0; 1) 2 B : then B = (B; _; ^; !; ; 0; 1) 2 K( D ), : c where for all x; y 2 B , x ! y = x _y : and x y = x ^ y c.
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De nition 15 If H = (H; :_; ^; !; 0; 1) 2 H : then H = (H; _; ^; !; ; 0; 1) 2 K( D ), : where for all x; y 2 H , x ! y = x ! y : and x y = :(x ! y). De nition 16 If B = (B; _; ^; ; 0; 1) 2 Br : : then B = (B; _; ^; !; ; 0; 1) 2 K( D ), : where for all x; y 2 B , x ! y = 1 (x y) : and x y = x y .
Remark 6 De ne the following three classes B = B = B 2 B , H = H = H 2 H and Br = fB = B 2 Brg. Then:
(i) B = H \ Br. (ii) B; H and Br are varieties of algebras termwise de nitionally equivalent with algebras respectively of B , H and Br. 33
:
:
Notation 3 If A = (A; _; ^; !; ; 0; 1) 2 K( D ) and x; y 2 A then we let: :
(1) (x) = 1 ! x, : (2) (x) = x 0, : : (3) :x = x ! 0, o (4) x ^ y = (x ^ y), _ (5) x _ y = (x _ y), (6) (A) = f (x) = x 2 Ag, (7) (A) = f (x) = x 2 Ag.
De nition 17 : Let D be the variety of all A = : (A; _; ^; !; ; 0; 1) 2 K( D ) satisfying:
(i) The reduct L(A) = (A; _; ^; 0; 1) is a bounded distributive lattice. (ii) For all x; y 2 A: : (d1) x ! (x _ y) = 1 (d2) (x _ y) = (x) _ (y) : (d3) (x ^ (x ! y)) = (x ^ y) : : : (d4) x ! (y ! z) = (x ^ y) ! z : : (d5) x ! y =: (x) ! y: : : (d6) x ! (y z) = 1 (x ^ (y ! z)) : (d7) x = (x) _ ( (x)^ :x) 34
o
:
(d1 ) (x ^ y) y = 0 (d2o) (x ^: y) = (x) ^ (y) (d3o) ((x: y): _ y) = : (x _ y) (d4o) (x : y) z: = x (y _ z) (d5o) x y = x: (y) : : : o (d6 ) (x ! y) z = :((x :y) _ z) (d7o) x = (x) ^ ( (x) _ (1 x)) The equations (d1o)-(d7o) are called dual of (d1) (d7). De nition 18 (i) An algebra A 2 D will be called a Brouwerian D-algebra (BrD-algebra). (ii) For any A 2 K( D ), de ne an algebra s e = (A; _ e;e e; ^ e ; !; A 0; e 1) 2 K( D ), where if e y = x ^ y; x ^ e y = x _ y; x x; y 2 A then x_ : s : ! y = y x; x e y = y ! x; e 0 = 1; e 1 = 0. (iii) If A 2 D then is the order relation on A corresponding to the lattice reduct L(A). The next lemma follows from previous de nitions. 35
e 2 D. Lemma 16 A 2 D implies A
Remark 7 A duality principle. Let LD be the language of BrD-algebras, V be the set of variables and T [LD ] be the set :of terms. Then : T[LD ] = (T [LD ]; _; ^; !; ; 0; 1) is an algebra in K( D ) with the free generating set s e D ] = (T [LD ]; _ e;e e; ^ e ; !; V . Let T[L 0; e 1) be the algebra of type D introduced in De nition 18 (ii). Then there exists a unique extension of the identitity function idV : V ! V to an isoe D ] . The next conmorphism o : T[LD ] ! T[L ditions hold, for all ; 2 T [LD ]: (1) o coincides with , if 2 V; (2) 0o (1o) is the con: stant: 1 (0); (3) ( _ )o; ( ^ )o; ( ! )o and ( ): o are respectively terms o ^ o; o _ : o o o ; and o ! o. A duality principle for D follows from Lemma 16: for all terms ; 2 T [LD ]; if the equation = is valid in D then the equation o = o is valid in D: 36
The next properties also follows from the above de nitions. On this basis the exact relation between BrD-algebras and the two kind of structures of Heyting algebra and dual Heyting algebra is established.
Corollary 17 If A 2 D then the following conditions hold: (i) is an interior operator of the poset (A; ; 0; 1). (ii) (A) is a : subalgebra of (A; _; !; 0; 1). (iii) (A) = : ( (A); _; ^o; !; 0; 1) is a Heyting algebra.
Corollary 18 If A 2 D then the following conditions hold: (i) is a closure operator of the poset (A; ; 0; 1):. (ii) the set (A) is a subalgebra of (A; ^; _ ; 0; 1): . (iii) the next system (A) = ( (A); _; ^; ; 0; 1) is a dual Heyting algebra. 37
Lemma 19 Let A 2 D and de ne a function i : A ! (A) (A) by i(x) = ( (x); (x)); for all x 2 A. Let (A) 2 H and (A) 2 Br be the systems associated with A by Corollary 17 (iii) and Corollary 18 (iii). The following conditions hold: (i) The algebra (A) :2 H is : de ned by (A) = ( (A); _; ^o; !; ; 0; 1). (ii) The algebra_ (A) 2: Br is de ned by : (A) = ( (A); _; ^; !; ; 0; 1). (iii) i is an injective homomorphism from A into (A) (A) such that the image algebra i(A) is a subdirect product of the pair ( (A); (A)). The following result is a model theoretical characterization of D. Theorem 20 D = H [ Br : Proof. We have H [ Br = \(V = V is a variety in K( D ) and H [ Br V). Using De nitions 1 and 15, respectively 3 and 16, from De nitions 17 and 18 (i) it follows that H [ Br D, thus H [ Br D. 38
Let V be any variety in K( D ) such that H [ Br V: We prove that in this case D V . Let A 2 D. Lemma 19 implies that A is isomorphic to a subalgebra of (A) (A) with (A) 2 H V and (A) 2 Br V , thus A 2 V . Therefore, the H [ Br . next inclusion also holds D The following lemma expresses the fact that any BrD-algebra includes a subposet which is a Boolean algebra. Then different basic properties of the variety D are presented. :
:
Lemma 21 Let A 2 D and :A = f:x = : x 2 Ag. Then: (i) :A = (A) \ (A): : (ii) The subposet (:A; ; 0; 1) of (A; ; 0; 1) is a Boolean lattice such that its corresponding Boolean algebra is de ned by B(A) = _ : : (:A; _; ^o;c ; 0; 1) 2 B , where if u 2 :A : c then its complement is de ned by u = :u: (iii) The algebra B(A) 2 B associated with B(A) 2 B by De nition 14 is given by the : _ : : system B(A) = (:A; _; ^o; !; ; 0; 1). 39
Theorem 22 The variety D is arithmetical i.e. D is congruence-distributive and congruencepermutable.
Theorem 23 The algebraic categories H; Br and B are re ective full subcategories of D such that their re ectors preserve the injective homomorphisms. It is known that injective Heyting (dual Heyting) algebras are just complete Boolean algebras. The following result shows that complete Boolean algebras of B are also injective algebras of D. Theorem 24 An algebra A 2 D is injective if and only if A 2 B and the reduct L(A) is a complete lattice. 40
The next result presents a factorization condition for BrD-algebras. Theorem 25 Let H ? Br be the class of all algebras A 2 K( D ) such that there exist a pair (H; B) 2 H Br and an isomorphism f 2 [A; H B]. Let A 2 D: The following conditions are equivalent: (i) A 2 H ? Br: (ii) There exists an element 2 A such that for all x 2 A: (1) (x ^ ) = x ^ and (2) (x _ ) = x _ :
6 Free BrD-algebras Theorem 26 Let X be a set, H(X) be a free algebra over H and Br(X) be a free algebra over Br with free generating set X . Let X = f(x; x) = x 2 Xg and A(X) 2 D be the subalgebra of the direct product H(X) Br(X) generated by X . Then A(X) is a free algebra over D with free generating set X . 41
De nition 19 A sentential language for D is a pair P = (P; K), where P is a set of atomic sentential :symbols and K is a set of binary : : _; ^; !; and unary ; ; : sentential connectives such that P \ K = ;.
De nition 20 The algebra of sentences is an : : : algebra F[P] = (F [P ]; _; ^; !; ; ; ; :) of type D = (2; 2; 2; 2; 1; 1; 1) with free generating set P .
De nition 21 For any A: 2 D, de ne an alge: : bra A = (A; _; ^; !; ; ; ; :) 2 K( D ), : where for all x 2 A; (x) = 1 ! x; (x) = : : : : x 0; :x = x ! 0 = 1 x. 42
De nition 22 Let A 2 D, f : P ! A, a 2 F [P ] and C D. Let f 2 [F[P]; A ] be the unique homomorphism in K( D ) extending f . The sentence a is called
(i) f -valid if f (a) = 1 and in this case we write f + a; (ii) A-valid if f + a, for all f : P ! A and in this case we write + A a; (iii) C -valid if + A a; for all A 2 C and in this case we write + C a; (iv) f -invalid if f (a) = 0 and in this case we write f a; (v) A-invalid if f a, for all f : P ! A and in this case we write A a; (vi) C -invalid if A a; for all A 2 C and in this case we write if C a: De nition 23 A sentential intuitionistic logical axiom of P is a sentence such that it has one of the following forms, with a; b; c 2 F [P ]: :
:
(1) a ! (b ! a) :
:
:
:
:
:
(2) (a ! (b ! c)) ! ((a ! b) ! (a ! c)) 43
:
:
:
(3) a ! (:a ! b) :
:
:
:
(4) (a ! :a) ! :a :
:
(5) a ! (b ! (a ^ b)) :
(6) (a ^ b) ! a :
(7) (a ^ b) ! b :
(8) a ! (a _ b) :
(9) b ! (a _ b) :
:
:
:
:
(10) (a ! c) ! ((b ! c) ! ((a _ b) ! c)) De nition 24 A sentential dual intuitionistic logical axiom of P is a sentence such that it has one of the next forms, with a; b; c 2 F [P ]: o
(1 ) (a o
(2 ) (3o) (4o) (5o) (6o) (7o)
:
:
b) a : : : ((c a) (b a)) : : : (b :a) a : : : : :a (:a a) : : ((a _ b) b) a : a (a _ b) : b (a _ b) 44
:
((c
:
b)
:
a)
o
(8 ) (a ^ b)
:
a (9o) (a ^ b) b : o (10 ) ((c (a ^ b)) :
:
(c
:
b))
:
(c
:
a)
De nition 25 A substitution rule of P is dened by one of the following relations, with a; b 2 F [P ]: (a) : : (b!b)!a (a) :
:
a (b b)
:
(a b) : : :(a!b) : (a!b)
:
:
:(a b)
:
a!b : (a)!b : a b : a (b)
:
:a : : a!(b b) : :a : : (b!b) a
De nition 26 An intuitionistic logical law is a sentence ' 2 F [P ] such that there exists a formal proof of ' from sentential intuitionistic logical axioms (De nition 23), substitution rules (De nition 25) and the rule of inference : a;a!b and in this case we write `+ ': b 45
De nition 27 A dual intuitionistic logical law is a sentence ' 2 F [P ] such that there exists a formal proof of ' from sentential dual intuitionistic logical axioms (De nition 24), substitution rules (De nition 25) and the rule of : inference b ba;a and in this case we write ` ': Free algebras over H (Br) are the Lindenbaum-Tarski algebras of sentential intuitionistic logic (dual intuitionistic logic). On this basis a construction of free objects over D is presented in the following theorem.
Theorem 27 Let be a binary relation such that for all formulas '; 2 F [P ], (eq) ': : : iff `:+ (' ! ) ^ ( ! ') and ` (' )_ ( '). Then the following conditions hold: (i) The relation
is a congruence of F[P]:
(ii) There exists a structure: of BrD-algebra : A(P ) = (A(P ); _; ^; !; ; 0; 1) 2 D with A(P ) = F [P ]= such that A(P ) is a free object over D with free generating set P . 46
Remark 8 Using the structure of free Heyting algebra with one free generator given by Nishimura ([1]; p:182 185) and Theorem 26 one can obtain a concrete construction of an algebra A(a) such that it is a free algebra over D with free generating set fag. Then A(a) 2 = H [ Br and the Boolean center of A(a) is the two elements set f0; 1g: Theorem 25 implies A(a) 2 = H ? Br. This property shows that the class D of all BrD-algebras is a proper extension of H ? Br with H [ Br $ H ? Br $ D.
7 Modal S2G-algebras and concluding remarks The previous results can be considered in the study of a special algebraic structure called modal S2G-algebra ( S2G-algebra). This structure combines S2G-algebra and BrDalgebra. A S2G-algebra is an S2G-algebra endowed with special modal operators. 47
In order to have an algebraic semantics for a suitable formal logic of two criteria decision making a S2G-algebra is de ned by a triple (A; ; ), where A = (A; _; ^; !; ; ; 0; 1) is an S2G-algebra and ; are operators such that the following conditions hold: The function : A ! A is an interior operator of the lattice reduct L(A) = (A; _; ^; 0; 1) and for all x; y 2 A; ( (x) ! (y)) = (x) ! (y): The function : A ! A is a closure operator of L(A) and for all x; y 2 A; ( (x) (y)) = (x) (y), where u v = (v ! u ) if u; v 2 A: :
:
The system A; _; ^; !; ;d ; 0; 1 is an involutive Brouwerian D-algebra, where for all x; y 2 A; the basic operations :are : de ned by x ! y = (x) ! (y); x y = (x) (y) and xd = (x ). 48
Basic properties of S2G-algebras follows from previous results. The main fact is that a S2G-algebra is the algebraic counterpart of a sentential modal logic of two criteria decision making ( DM2-logic). A basic aspect of interest with respect to MathFuzzLog group activity is the connection between multi-criteria decision making and mathematical fuzzy logic. There are different theoretical and practical interests regarding the study of this connection: Elaboration of the theory of algebras and their logic
S2 G -
Multisymmetric Gödel algebras (S Galgebras) Mathematical logic of multiple criteria decision making and fuzzy logic 49
Remark 9 A real universe for multisymmetric Gödel algebras is R ( an ordinal number).
R = R [ f 1; +1g ( 1; +1 2 = R) THANK YOU !
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