Characterizing classes defined without Equality

Characterizing classes defined without Equality R. Elgueta∗ Departament de Matem` atica Aplicada II Universitat Polit`ecnica de Catalunya Pau Gargallo, 5 08028 Barcelona, Spain ∗

Abstract.

In this paper we mainly deal with first-order languages without equality and introduce a weak form of equality predicate, the so-called Leibniz equality. This equality is characterized algebraically by means of a natural concept of congruence; in any structure, it turns out to be the maximum congruence of the structure. We show that first-order logic without equality has two distinct complete semantics (full semantics and reduced semantics) related by a reduction operator. The last and main part of the paper contains a series of Birkhoff-style theorems characterizing certain classes of structures defined without equality, not only full classes but also reduced ones.

1. Preliminaries 1.1. Basic Notation and Terminology. Let the triple L = hF, R, ρi be a first order language; F and R denote pairwise disjoint sets of function and relation symbols of L respectively (R must be nonempty), and ρ is the arity function from F ∪ R into the set of natural numbers. We use capital Gothic letters A, B, C, . . . , with appropriate subscripts, to represent structures over L, also called L-structures. In order to be consistent with the notation, we denote by A the universe of A, and by FA and RA the interpretations on A of the collections of function and relation symbols of L respectively, i.e., FA = {f A : f ∈ F } and RA = {rA : r ∈ R}. The corresponding boldface letter A is used to denote the underlying algebra hA, FA i of A, and we normally write f A instead of f A . Lowercase boldface letters a,b,. . . are used to indicate members of the cartesian product of some family of sets. So, if A is an L-structure, a = ha1 , . . . , an i belongs to An , f ∈ F and r ∈ R, and h is any mapping with domain A, then f A a, a ∈ rA and ha are short-hand for f A a1 . . . an , ha1 , . . . , an i ∈ rA and hha1 , . . . , han i, respectively. By an L-algebra we mean the underlying algebra of any L-structure; of course, if the set of function symbols is empty, an L-algebra simply means an arbitrary ∗ Work partially supported by grant EE92/2-260 of CIRIT, Generalitat de Catalunya. E-mail address: [email protected].

Typeset by AMS-TEX 1

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set. The absolutely free L-algebra over a set of ω variables, i.e., the algebra of all L-terms over ω variables, is denoted by TeL . Formulas are represented by lowercase greek letters ϕ, ψ, ϑ, . . . , and uppercase ones are used to denote sets of formulas. We write ϕ(x1 , . . . , xn ) to indicate that the free variables that occur in ϕ are among x1 , . . . , xn . Given an L-structure A, an algebra homomorphism g : TeL →A (called assignment) and an L-formula ϕ, we use the notation A |= ϕ[g] to indicate that the formula ϕ holds in the structure A under the assignment g, in the usual way. Following the standard convention, A |= ϕ expresses that A satisfies the universal closure of ϕ. When we write A |= ϕ(x1 , . . . , xk ) [a1 , . . . , ak ], we mean A satisfies ϕ with respect to any assignment g : TeL →A such that gxi = ai , for all 1 ≤ i ≤ k. We say L is a language with equality, or simply L has equality, when L contains a binary relation symbol ≈ which is always interpreted as the identity; in other terms, only the structures A for which ≈A is the diagonal relation, i.e., the set ∆A = {ha, ai : a ∈ A}, count as L-structures. On the contrary, we say L is a language without equality, or L has no equality, provided that L does not contain any such binary relation symbol. Thus, if L is without equality and r is some binary relation symbol of L, then r can be interpreted in the L-structures as any binary relation whatsoever∗∗ . 1.2. Substructures, Filter Extensions and Elementary Substructures. If A = hA, RA i and B = hB, RB i are two L-structures, we say A is a substructure of B, in symbols A ⊆ B, if A is a subalgebra of B and rA = rB ∩ Aρ(r) for all r ∈ R. Likewise, B is a filter extension of A, and in this case we write A 4 B, provided the underlying algebras of A and B coincide and rA ⊆ rB for every r ∈ R. Given a subset X of A, we use the notation A ¹ X to understand the substructure of A generated by X, i.e., A ¹ X = h[X], {f A ¹ [X] : f ∈ F }, {rA ∩ [X]ρ(r) : r ∈ R}i, where [X] denotes the universe of the subalgebra of A generated by X. We call A an elementary substructure of B, in symbols A ⊆e B, iff A ⊆ B and for any L-formula ϕ and any assignment g of elements of A to L-terms, the equivalence A |= ϕ [g] iff B |= ϕ [g] holds. The basic logical relation between structures is provided by the notion of elementary equivalence, which is a weakening of the preceding condition; recall from classical model theory that two structures A, B over L are elementarily equivalent iff every L-sentence true in A is also true in B, and viceversa. Obviously, A ⊆e B implies A ≡ B. ∗∗ Usually the distinction between a language with or without equality relies on the presence or not of the equality symbol among the logical constants. We do not follow this widely accepted convention in the preceding definition; the reason is that in this way all the results we state in the sequel amount to well-known results in the case the language has equality.

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1.3. Homomorphisms. A mapping h : A → B is said to be a homomorphism from A into B if h is an algebra homomorphism from A into B and the condition (1)

ha1 , . . . , an i ∈ rA =⇒ hha1 , . . . , han i ∈ rB .

holds for all n-ary relation symbol r ∈ R and all a1 , . . . , an ∈ A; we write h : A → B to indicate that h is such a homomorphism. We say that h is an embedding or an epimorphism provided it is as a homomorphism between the underlying algebras; in these cases we put h : A ½ B and h : A ³ B respectively. When h is onto we also say that B is a homomorphic image of A. Finally, h is an isomorphism between A ∼ B, if h is one-one and onto and the inverse of h is also and B, in symbols h : A = a homomorphism. We call h : A → B a strong homomorphism from A into B, and we write h : A →s B, if h is a homomorphism from A into B for which the reverse implication of (1) also holds; so for all n-ary relation symbols r ∈ R and all a1 , . . . , an ∈ A, (2)

ha1 , . . . , an i ∈ rA ⇐⇒ hha1 , . . . , han i ∈ rB .

Strong homomorphisms that are one-one are called strong embeddings, whereas those that are surjective are referred to as reductive homomorphisms; we write, respectively, h : A ½s B and h : A ³s B. If there is a reductive homomorphism from A onto B we say that B is a reduction of A and A an expansion of B. Note that a bijective strong homomorphism is simply an isomorphism as it is defined before and that reductive homomorphisms are the same as isomorphisms when the language has equality. Both assertions are easy consequences of the following result. Lemma 1.1. The following holds for every algebra homomorphism h : A →B. (i) h : A → B iff rA ⊆ h−1 rB , for all r ∈ R. (ii) h : A →s B iff rA = h−1 rB , for all r ∈ R. (iii) h : A ³s B implies rA = h−1 rB and hrA = rB , for all r ∈ R.  For each h : A → B, we define the image of A by h as the structure hA = hhA, hRA i, where hRA = {hrA : r ∈ R} and hrA = {ha ∈ B n : a ∈ rA }. Conversely, we define h−1 B = hh−1 B, h−1 RB i, where h−1 RB = {h−1 rB : r ∈ R} and h−1 rB = {a ∈ An : ha ∈ rB }, and call h−1 B the preimage of B by h. Both hA and h−1 B are again structures over L, even though hA is not in general a substructure of B nor h−1 B a substructure of A. This is true, however, when h is a strong homomorphism. The next proposition states a generalized form of this property. Lemma 1.2. Let h : A →s B. For each substructure A0 of A we have that hA0 ⊆ B. Conversely, if B0 is a substructure of B then h−1 B0 ⊆ A.  Observe that every surjective homomorphism from A onto B can be canonically decomposed through a reductive homomorphism. Concretely, if h : A ³ B, then h maps strong homomorphically h−1 B onto B (h−1 B is in fact the least filter extension of A satisfying this property!). So, the composition of the identity id :

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ˆ : h−1 B ³s B (ha ˆ = ha) coincides with h : A ³ B. As a result, A ³h−1 B and h every homomorphism h : A → B factorizes according to the following diagram: A   idy

(†)

h

−−−−→ B x j 

h−1 B −−−−→ hA ˆ h

This decomposition is essential because, as we shall see later on, the role homomorphisms play in universal algebra are performed by strong homomorphisms when we deal with languages that contain some relation symbol. A homomorphism h : A →B is said to be elementary, in symbols h : A →e B, iff for any L-formula ϕ and any assignment g, A |= ϕ [g] iff B |= ϕ [h ◦ g]. Evidently, if h : A →e B then A ≡ B. The next proposition will be used several times in the sequel; its converse is not true but a weaker implication is contained in Corollary 4.6 below. Proposition 1.3. Every reductive homomorphism is elementary. Proof. Let h be a reductive homomorphism from A onto B. We claim that A |= ϕ [g] iff B |= ϕ [h ◦ g] for every formula ϕ over L and every assignment g. The proof goes by induction on the logical complexity of ϕ. Clearly the statement is true if ϕ is an atomic formula. Moreover, the induction step is obvious when ϕ is a negation or a conjunction. Hence, suppose that ϕ is ∃xψ(x) for some formula ψ. Then A |= ∃xψ(x) [g] iff A |= ψ(x) [g(a/x)] for some a ∈ A, which is equivalent, by the induction hypothesis, to the condition that B |= ψ(x) [h ◦ g(a/x)] for some a ∈ A. But B |= ψ(x) [h ◦ g(a/x)] iff B |= ψ(x) [(h ◦ g)(ha/x)]. Thus, the fact that h is surjective completes the proof.



1.4. Product Constructions. Assume that Ai = hAi , RAi i, with i ∈ I, is a family of L-structures. We define the direct product of {Ai : i ∈ I} by setting Q i∈I

Q Q Ai := h i∈I Ai , i∈I RAi i,

Q where Q i∈I Ai is the usual direct product of the Q underlying algebras {Ai : i ∈ I} and i∈I RAi denotes the interpretations on i∈I Ai of the symbols of R defined in the obvious way: if r ∈ R is an n-ary relation symbol and aji means the ith component of aj , for each 1 ≤ j ≤ n, r

Q i∈I

Ai

:= {ha1 , . . . , an i ∈

¡Q i∈I

Ai

¢n

: ha1i , . . . , ani i ∈ rAi for all i ∈ I}.

Q We allow I to be empty; in this case, i∈I Ai is the trivial structure with a one element underlying algebra and where all relations are non-empty.

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The direct product construction can be extended in several ways, three of which are useful for our purposes. The first one is the notion of filtered product. Suppose that F is a proper filter of Sb(I) and define ΘF := {ha,bi ∈

¡Q

Ai

i∈I

¢2

: {i ∈ I : ai = bi } ∈ F}.

As is well known, ΘF is a congruence relation on the algebra can put Q i∈I

where

QF

QF i∈I

i∈I

Ai /F :=

Q i∈I

Ai /ΘF ,

Q i∈I

RAi /F :=

Q

QF i∈I

i∈I

Ai , so that we

RAi /ΘF ,

RAi denotes the family of relations

rAi := {ha1 , . . . , an i ∈ (

Q i∈I

Ai )n : {i ∈ I : ha1i , . . . , ani i ∈ rAi } ∈ F},

QF for r ∈ R (actually, for each r, the set i∈I rAi is just the least relation that contains Q r i∈I Ai and is compatible with ΘF ). Then, the filtered product of {Ai : i ∈ I} by F is defined as the structure Q i∈I

Q Q Ai /F := h i∈I Ai /F, i∈I RAi /Fi,

which Q coincides with the direct product in the case F = {I}. For simplicity, if a ∈ i∈I Ai , the equivalence class of a modulo ΘF is denoted by a/F. We point out that the projection map from the direct product either onto its components or onto the quotient modulo ΘF is not, in Q homomorQ general, a strong A onto phism. Actually, if π means the projection from i F i∈I Ai /F, the i∈I Q preimage of i∈I Ai /F by πF is QF i∈I

QF Q Ai = h i∈I Ai , i∈I RAi i.

Q It is easy to show that, if L has equality, the filtered product i∈I Ai /F Q is again an L-structure for any proper filter F, i.e., the interpretation of ≈ on i∈I Ai /F is again the identity relation. This is in fact a consequence of the following result, whose proof can be found in almost every model theory textbook; see, e.g., [4]. Theorem 1.4. Let ϕ = ϕ(x1 , . . . , xn ) be an arbitrary Horn formula and let Ai , i ∈ I, be a family of L-structures over a nonempty index set I. Let F be a proper Q filter of Sb(I). If g : TeL → i∈I Ai is any assignment, then {i ∈ I : Ai |= ϕ [πi ◦ g]} ∈ F implies

Q i∈I

Ai /F |= ϕ [πF ◦ g].

Moreover, this implication becomes an equivalence if ϕ is atomic.  Q A filtered product i∈I Ai /U is called an ultraproduct if U is an ultrafilter on the index set I. The following is a well-known property of ultraproducts; see also [4].

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Theorem 1.5. (ÃLos Theorem) Let I be a nonempty set. AssumeQAi , i ∈ I, are L-structures and let U be an ultrafilter of Sb(I). If g : TeL → i∈I Ai and ϕ = ϕ(x1 , . . . , xn ) is an arbitrary first-order formula over L, then {i ∈ I : Ai |= ϕ [πi ◦ g]} ∈ U iff

Q i∈I

Ai /U |= ϕ [πU ◦ g]. 

The second generalization of the direct product construction comes from universal algebra. A structure A is called a subdirect product Q Q of the system {Ai : i ∈ I}, in symbols A ⊆sd i∈I Ai , if A is a substructure of i∈I Ai and the restriction Qof the projection map πi to A Q is surjective for every i ∈ I. An embedding h : A ½ i∈I Ai Q is subdirect if hA ⊆sd i∈I Ai ; we write h : A ½sd i∈I Ai to mean that h is a subdirect embedding. Note that Q every subdirect embedding is strong; indeed, if Q h : A ½sd i∈I Ai , then hrA = r i∈I Ai ∩Q(hA)n for every n-ary relation symbol r. So, h being one-one, hha1 , . . . , han i ∈ r i∈I Ai implies ha1 , . . . , an i ∈ rA , for all a1 , . . . , an ∈ A. Finally, a last generalization of direct products that combines filtered and subdirect products has been recently introduced by Czelakowski [7]; we also use it in subsequent sections. Let A be a subdirect product of a system {Ai : i ∈ I} of L-structures, and let F be a proper filter on I. It is an easy matter to check that the restriction of ΘF to A, ΘF ,A := ΘF ∩ A2 , is a congruence on the algebra A. So we define the filtered subdirect product of A by F as the structure F A/F := hA/ΘF ,A , RA /ΘF ,A i, F where RA := {rA,F : r ∈ R} and, for all r ∈ R,

rA,F :=

QF i∈I

rAi ∩ Aρ(r) .

Let us notice that, Q by the definition, A/F is isomorphic to a substructure of the filtered productQ i∈I Ai /F, because in fact the quotient algebra A/ΘF ,A can be embedded into i∈I Ai /F; the embedding is established by the mapping that assigns the equivalence class a/F of a (modulo ΘF ) to the element a/ΘF,A , for all a/ΘF ,A ∈ A/ΘF ,A . Also, observe that the projection from A to A/F given by F a 7−→ a/ΘF ,A need not be strong; the preimage of A is the structure AF := hA, RA i.

2. A weak predicate of equality The idea of defining the identity relation in second-order logic as x ≈ y ↔ ∀P (P (x) ↔ P (y)), where P is a variable ranging over all predicates, goes back to Leibniz. This definition has a natural first-order counterpart, which constitutes a weak form of equality. Namely, if A is an L-structure and a, b are two members of its domain, a and b are related if the condition A |= ϕ(x) ↔ ϕ(y) [a, b]

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holds for every first-order formula ϕ of L with at least one free variable. In this section we introduce the concept of congruence on a structure and show how it is related to this new predicate of equality, which is called equality in the sense of Leibniz, or simply Leibniz equality. 2.1. Congruences on a structure. Let A = hA, RA i be an L-structure. A binary relation θ on A is said to be a congruence on A if θ is a congruence on the underlying algebra which is compatible with all the relations belonging to RA , i.e., for every r ∈ R of arity n, ha1 , . . . , an i ∈ rA and ai ≡ bi (θ), for 1 ≤ i ≤ n =⇒ hb1 , . . . , bn i ∈ rA . For simplicity, if a = ha1 , . . . , an i, b = hb1 , . . . , bn i ∈ An we write a ≡ b (θ) to mean that ai ≡ bi (θ) for all 1 ≤ i ≤ n. The compatibility property of θ with an arbitary n-subset D of A can then be expressed as follows: if a ∈ D then b ∈ D for every b ∈ An such that a ≡ b (θ). In this case, D=

S

a∈D (a1 /θ)

× (a2 /θ) × . . . × (an /θ).

Clearly the set of all congruences on A, denoted Co A, is a subset of Co A. The following proposition provides a full description of this subset. Proposition 2.1. For every L-structure A, the poset Co A = hCo A, ⊆i is a principal ideal of Co A. Proof. Clearly φ ⊆ θ and θ ∈ Co A implies that φ ∈ Co A. Hence, it suffices to W show that for each family {θi : i ∈ I} of congruences on A, i∈I θi belongs to Co A. Let a=Wha1 , . . . , an i, b = hb1 , . . . , bn i ∈ An . From universal algebra, we know a ≡ b ( i∈I θi ) iff there exists a sequence of elements c1 , . . . , ck ∈ An and i1 , . . . , ik−1 ∈ I such that a = c1 , ck = b and cj ≡ cj+1 (θij ), Thus, the compatibility of

W i∈I

1 ≤ j ≤ k − 1.

θi follows immediately.



Extending the notation introduced in [1], we denote by ΩA the largest congruence on A. So, by the previous lemma, Co A = {θ ∈ Co A : θ ⊆ ΩA}. Notice that ΩA 6= ∇A whenever rA 6= Aρ(r) and rA 6= ∅ for some relation symbol r ∈ R (∇A denotes the set of all ordered pairs of members of A). Also, if A contains a binary relation that satisfies the axioms of equality, ΩA coincides with this relation. In fact, a structure A is said to be reduced if Co A = {∆A } or, equivalently, if ΩA = ∆A . So if L has equality, then any L-structure is reduced. Examples. Let L be a language with possibly some function symbols and a sole relation symbol, of arity 2. Assume A is an L-algebra. We claim that a congruence

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φ on A is compatible with a binary relation R on A iff φ · R · φ ⊆ R, where · denotes the relative product of any two binary relations; the proof is straightforward and is provided for instance in [2, Prop. 5.7]. Using this fact it is easy to conclude the following: for any binary relation θ on A, the largest congruence of A = hA, θi is W (1) {φ ∈ Co A : φ ⊆ θ}, if θ is an equivalence relation on A; W (2) {φ ∈ Co A : φ · θ · φ ⊆ θ}, if θ is a reflexive, symmetric relation compatible with the functions on A, i.e., such that for any function symbol f and any a, b ∈ Aρ(f ) , f A a ≡ f A b (θ) whenever a ≡ b (θ); (3) θ∩θ−1 , if θ is a quasi-order on A, i.e., a reflexive, transitive relation compatible with the functions on A; (4) θ, if θ is a congruence on A. There exists a useful external (categorical) characterization of the notion of congruence on a structure based on the concept of kernel. Namely, for any homomorphism h : A →B, let us define the kernel of h as the set Ker h = h−1 ∆B . Then the next result contains part of this characterization, which is completed by Proposition 3.1 below. Lemma 2.2. Let A, B be two L-structures and let h : A →B. If h : A →s B then Ker h ∈ Co A. Conversely, if Ker h ∈ Co A and rB = hrA for all r ∈ R, then h : A →s B. Proof. Clearly Ker h ∈ Co A. Let r be any relation symbol and let a, a0 ∈ Aρ(r) be such that a ∈ rA and a ≡ a0 (Ker h). Since h is strong, ha0 = ha ∈ rB implies a0 ∈ rA , and so Ker h belongs to Co A. Assume now that Ker h ∈ Co A and rB = hrA for all r ∈ R. If ha ∈ rB , there is an element a0 ∈ rA such that ha = ha0 . Thus a ≡ a0 (Ker h) and, consequently, a ∈ rA is equivalent to ha ∈ rB .  Some natural questions concerning the relation between the lattice of congruences of certain structures and those of their substructures, homomorphic images and products arise. We shall not enter into this subject, but merely state two results that tell us something in this sense and that will become useful later on. Lemma 2.3. Let A be an L-structure and let B ⊆ A. For every binary relation θ ⊆ A2 , define θB = θ ∩ B 2 . Then, θ ∈ Co A implies θB ∈ Co B. Proof. Clearly θB is a congruence on the underlying algebra of B. The fact that θB is compatible with the relations of B follows directly from the definition of substructure.  Lemma 2.4. For all h : A →s B, φ ∈ Co B implies h−1 φ ∈ Co A. If, moreover, h is a reductive homomorphism, then θ ∈ Co A and θ ⊇ Ker h implies hθ ∈ Co B. Proof. Suppose that h is strong and φ is a congruence on B. Obviously, h−1 φ is an equivalence relation on A. Let a = ha1 , . . . , an i, a0 = ha01 , . . . , a0n i ∈ An be such that a ≡ a0 (h−1 φ). Then ha ≡ ha0 (φ), so that f B ha1 , . . . , han ≡ f B ha01 . . . ha0n (φ) for all n-ary function symbol f and, consequently, f A a1 , . . . , an ≡ f A a01 . . . a0n (h−1 φ). Hence, h−1 φ is a congruence on A. Moreover, if r is an n-ary relation symbol, a ∈ rA

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and a ≡ a0 (h−1 φ) imply ha ∈ rB and ha ≡ ha0 (φ). Then, since φ is compatible with relations, ha0 ∈ rB , which entails a0 ∈ rA as h is strong. As a result, h−1 φ is compatible with relations and hence a congruence on A. The proof of the converse is also a straighforward consequence from the assumptions. Now the fact that h is surjective and Ker h ⊆ θ is used to show that hθ is still an equivalence relation. Let us see that hθ is transitive as example. Take b1 , b2 , b3 ∈ B such that b1 ≡hθ b2 ≡hθ b3 . Then ha1 = b1 , ha2 = b2 = ha02 and ha3 = b3 for some a1 , a2 , a02 , a3 ∈ A satisfying ha1 , a2 i, ha02 , a3 i ∈ θ; hence, as ha2 , a02 i ∈ Ker k ⊆ θ, we have a1 ≡ a3 (θ) and consequently hb1 , b3 i ∈ hθ. So hθ is transitive. To prove that hθ is a congruence, suppose b = hb1 , . . . , bn i, b0 = hb0 1 , . . . , b0 n i ∈ B n are such that b ≡ b0 (hθ). This means that for all 1 ≤ i ≤ n there exist ai , a0i ∈ A satisfying ai ≡ a0i (θ),

hai = bi ,

ha0i = b0i .

And from here we conclude the desired compatibility condition of hθ with the functions and relations of B.  The following is an interesting consequence from the last lemma. Theorem 2.5. Let A, B be two L-structures. If h : A ³s B, the following holds. (i) h−1 ΩB = ΩA; (ii) hΩA = ΩB. Proof. Evidently h−1 ΩB ∈ Co A. Assume θ ∈ Co A and let θ0 = θ ∨ Ker h, where the supremum is taken in the lattice Co A. By Lemma 2.4, hθ0 ∈ Co B, so that hθ ⊆ ΩB. Hence, θ ⊆ h−1 hθ ⊆ h−1 ΩB, which proves that h−1 ΩB is the greatest congruence on A. On the other hand, since h is surjective, φ ∈ Co B implies φ = hh−1 φ ⊆ hΩA. Thus, using 2.4, (ii) is also proved.  2.2. The Leibniz equality. Our next goal is to see that the largest congruence of a structure is just the algebraic counterpart of a purely logical concept, viz. the concept of equality in the sense of Leibniz that we announced earlier. To this end, we define a Leibniz formula over L to be any formula ψ(x, y) with two free variables such that, for some atomic L-formula ϕ = ϕ(x, z1 , . . . , zk ) with at least one free variable x, ψ(x, y) := ∀z1 . . . ∀zk (ϕ(x, z1 , . . . , zk ) ↔ ϕ(y, z1 , . . . , zk )). Then we have the next fundamental result. Theorem 2.6. If A is an L-structure, then a ≡ b (ΩA) iff A |= ψ(x, y) [a, b] for all a, b ∈ A and all Leibniz formulas ψ(x, y) over L. Proof. Let θ be the set of all pairs ha, bi such that A |= ψ(x, y) [a, b] for all Leibniz formulas ψ. One easily verifies that θ is an equivalence relation. In order to see that

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it is actually a congruence, let f be any n-ary function symbol. We have to show that, if a ≡ b (θ), where a, b ∈ An , then f A a ≡ f A b (θ). Since θ is transitive, it suffices to prove the condition (3)

f A b1 . . . bi−1 ai ai+1 . . . an ≡ f A b1 . . . bi−1 bi ai+1 . . . an (θ),

for all i ≥ 1. Let ψ(x, y) be any Leibniz formula and select any pairwise distinct variables w1 , . . . , wn−1 not in ψ. Let ϑ be the formula that results of simultaneously substituting f w1 . . . wi−1 xwi . . . wn−1 for x and f w1 . . . wi−1 ywi . . . wn−1 for y in ψ(x, y). Then A |= ψ(x, y) [f A b1 . . . bi−1 ai ai+1 . . . an , f A b1 . . . bi−1 bi ai+1 . . . an ]

iff

A |= ϑ(x, y, w1 , . . . , wn−1 ) [ai , bi , b1 , . . . , bi−1 , ai+1 , . . . , an ]. Hence, since ∀w1 . . . ∀wn−1 ϑ(x, y, w1 , . . . , wn−1 ) is again a Leibniz formula over L and ai ≡ bi (θ), the second condition holds and (3) is proved. Assume now that r is an n-ary relation symbol of L and that a ∈ rA , a ≡ b (θ) hold for some members a = ha1 , . . . , an i, b = hb1 , . . . , bn i of An . Take ϕ to be the atomic formula rz1 . . . zi−1 xzi . . . zn−1 . Then A |= ψ(x, y) [ai , bi ] and, consequently, we have the equivalence hb1 . . . bi−1 ai ai+1 . . . an i ∈ rA iff hb1 . . . bi−1 bi ai+1 . . . an i ∈ rA . This is true for all i ≥ 1, so a ∈ rA implies b ∈ rA , and θ is a congruence on A. Finally, suppose Φ is another congruence on A, a ≡ b (Φ) and c1 , . . . , ck ∈ A. If t1 , . . . , tn are terms over L whose free variables are among x, z1 , . . . , zk then A tA i (a, c1 , . . . , ck ) ≡ ti (b, c1 , . . . , ck ) (Φ),

for 1 ≤ i ≤ n.

Thus, if r is any n-ary relation symbol, the compatibility of Φ with relations implies A |= rt1 (x,z) . . . tn (x,z) ↔ rt1 (y,z) . . . tn (y,z) [a, b, c1 , . . . , ck ], where z = hz1 , . . . , zk i. As a result, A |= ψ(x, y) [a, b] for each Leibniz formula ψ and a ≡ b(θ). This shows that θ is the greatest element of Co A and completes the proof.  An easy induction on the complexity of the formulas allows us to prove that the atomic predicate ϕ in the Leibniz formulas can be replaced by arbitrary elementary predicates. Therefore, we actually have the following logical description of the largest congruence of a structure. Corollary 2.7. Let A be an L-structure and let a, b ∈ A. Then a ≡ b (ΩA) iff for any first-order formula ϕ := ϕ(x, z1 , . . . , zk ) over L and any c1 , . . . , ck ∈ A, A |= ϕ(x, z1 , . . . , zk ) [a, c1 , . . . , ck ] iff A |= ϕ(x, z1 , . . . , zk ) [b, c1 , . . . , ck ].  The previous result shows that a congruence on an L-structure identifies some of the elements of its domain whenever they satisfy exactly the same set of elementary properties that can be expressed in L. In particular, this identification is carried

11

as far as possible by the largest congruence: in this case, any two elements are related if and only iff they are “equal” in the above sense, i.e., they can be mutually replaced in any elementary predicate with no change on truth. Because of this, we shall use indistinctly the expressions Leibniz congruence of A and Leibniz equality in A to mean the relation ΩA. We refer the reader to [12] for some results concerning the properties of the Leibniz equality in a class of structures.

3. Full and Reduced Classes In this section we introduce the concept of quotient structure and state both reduced and nonreduced structures as building blocks of two distinct complete semantics for languages without equality. Some aspects of these two semantics are investigated in the last section. 3.1. Quotient Structures and Homomorphism Theorem. Let A be an Lstructure and θ a congruence on A. We construct a new L-structure A/θ on the quotient set A/θ = {a/θ : a ∈ A} as follows. For each n-ary function symbol f in F and each a1 , . . . , an ∈ A, we put f A/θ a1 /θ . . . an /θ = f A a1 . . . an /θ; similarly, for each n-ary relation symbol r in R, let n

rA/θ = {ha1 /θ, . . . , an /θi ∈ (A/θ) : ha1 , . . . , an i ∈ rA }. Evidently, the interpretations of the symbols as defined above do not depend on the choosen representatives, since θ is compatible with functions and relations of A. Thus A/θ is well defined; it is called the quotient of A modulo θ. From now on, the notations FA/θ and RA/θ will mean the interpretations of the symbols of F and R in A/θ respectively. The following result provides a converse of Lemma 2.2; it shows that every congruence is the kernel of a reductive homomorphism and thus completes the external characterization of congruences on a structure announced before. Proposition 3.1. Assume A is an L-structure and θ ∈ Co A. Then the natural mapping πθ from A into A/θ given by πθ a = a/θ is a reductive homomorphism such that Ker πθ = θ.  The last proposition can also be used to conclude that quotient structures are reductions. The converse is true again and allows us to state a homomorphism theorem that extends the classical Homomorphism Theorem of universal algebra. We base its proof on a new lemma that is important by itself. Lemma 3.2. Let A, B, C be L-structures and assume that h : A →B and g : A ³s C satisfy Ker g ⊆ Ker h. Then there exists a homomorphism k : C →B such that h = k ◦ g. Moreover, h is strong iff k is strong.

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Proof. Given c ∈ C, choose a ∈ A such that g(a) = c and define k(c) = h(a). The condition Ker g ⊆ Ker h says that k is an algebra homomorphism from C into B. Indeed, let c1 , . . . , cn ∈ C and a, a1 , . . . , an ∈ A satisfying that g(a) = f C c1 . . . cn and g(ai ) = ci , 1 ≤ i ≤ n. Then ha, f A a1 . . . an i ∈ Ker g ⊆ Ker h, so that k is an algebra homomorphism. Finally, if c = hc1 , . . . , cn i ∈ rC for some n-ary relation symbol r ∈ R, we have that a = ha1 , . . . , an i ∈ rA and hence ha = kc ∈ rB . If, in addition, h is strong, then kc ∈ rB implies a ∈ rA and consequently c ∈ rC .  Theorem 3.3. (Homomorphism Theorem) Given any two L-structures A and B, if h : A ³s B then A/Ker h ∼ = B. Proof. The proof is a straighforward consequence of Proposition 3.1 and the preceding lemma.  The reader should notice that there are also easy versions of the remaining Isomorphisms Theorems, including the Correspondence Theorem. 3.2. Leibniz Quotient of a Structure. For any L-structure A = hA, RA i, the quotient of A modulo ΩA is called the Leibniz quotient of A. For simplicity, we write A∗ to mean A/ΩA; A∗ denotes the underlying algebra of A∗ and a∗ is sometimes used to represent the equivalence class a/ΩA, for each a ∈ A. Given two L-structures A, B and a mapping h : A →B, we denote by h∗ the correspondence a∗ 7−→ (ha)∗ induced by h between the quotient sets A∗ and B ∗ ; it is not in general a well-defined mapping. By Proposition 3.1, A∗ is a reduction of A. Moreover, A∗ is a reduced structure by the Correspondence Theorem mentioned two paragrahs above, so that A∗∗ ∼ = A∗ . ∗ The next results show that actually the Leibniz quotient A is minimal in the sense that it is a reduction of any other reduction of A. Proposition 3.4. For each h : A ³s B, the correspondence h∗ defines an isomorphism between A∗ and B∗ . More generally, if h : A ³B then h∗ : (h−1 B)∗ ∼ = B∗ . Proof. Assume h : A ³s B. By 2.5, a ≡ a0 (ΩA) iff ha ≡ ha0 (ΩB), so that h∗ is well defined and one-one. Moreover, for any a= ha1 , . . . , an i ∈ An , if a∗ = ha∗1 , . . . , a∗n i then ∗

∗

h∗ f A a∗ = (hf A a)∗ = (f B ha)∗ = f B h∗ a∗ . ∗

∗

Also, since h is strong, a∗ ∈ rA iff h∗ a∗ ∈ rB . Hence h∗ is an isomorphism. To see that h∗ : (h−1 B)∗ ∼ = B∗ if h is an onto homomorphism, it suffices to apply the decomposition stated in diagram (†).  Corollary 3.5. Let B an L-structure. If B is a reduction of A, the Leibniz quotient A∗ is a reduction of B. Also, if A ∼ = B, then A∗ ∼ = B∗ .  The importance of the Leibniz quotients rests on the fact that the Leibniz equality in them coincides with the common identity relation.

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3.3. Completeness Theorem. Let Γ be any set of L-formulas, and let M od Γ = {A : A |= ϕ for all ϕ ∈ Γ}, M od∗ Γ = {A ∈ M od Γ : A is reduced}. M od Γ and M od∗ Γ are called, respectively, the full model class and the reduced model class of Γ. Their relationship can be expressed as follows. If K is any class of L-structures, define L(K) = {A : A ∼ = B∗ for some B ∈ K} (for simplicity, we normally write K∗ to mean L(K)). Then, since A∗ is elementary equivalent to A by Corollary 1.3, we have that M od∗ Γ = L(M odΓ). The operator L is called reduction operator. If K is an arbitrary class of Lstructures, we say K is a full class whenever it is closed under expansions and reductions; also, we say that K is a reduced class if it is obtained by applying the reduction operator to some other arbitrary class. In particular, the whole class of reduced L-structures is called reduced semantics to differentiate it from the class of all the L-structures, named full semantics. Observe that, if every member A of K satisfies rA = ∅ or rA = Aρ(r) , for all r ∈ R, then K∗ consists of one-element algebras endowed with empty and/or all relations. Following the standard notation, for any set Σ of L-formulas and any single L-formula ϕ, we write Σ |= ϕ to indicate that for all L-structures A and all assignments g, A |= ϕ [g] holds whenever A |= Σ [g] holds. Similarly, Σ |=∗ ϕ will indicate that for any reduced L-structure A and any assignment g, A |= Σ [g] implies A |= ϕ [g]. Then we have the following theorem. Theorem 3.6. (Completeness Theorem) Let Σ be a set of first-order L-sentences and ϕ a single first-order L-sentence. Then Σ ` ϕ iff Σ |= ϕ iff Σ |=∗ ϕ. Proof. The first equivalence is just the contents of G¨odel’s Completeness Theorem, and the second one is a direct consequence of Proposition 1.3. 

4. Fundamental Lemmas 4.1. Some Properties of Class Operators. In order to investigate the algebraic properties of classes of structures, let us introduce the operators that correspond to the constructions defined so far and let us state some technical lemmas. For any class K of L-structures, define S(K) = {A : A ∼ = C and C ⊆ B for some B ∈ K}, Se (K) = {A : A ∼ = C and C ⊆e B for some B ∈ K}, ∼ F (K) = {A : A = C and B4C for some B ∈ K}, H(K) = {A : A ∼ = C and h : B ³C for some B ∈ K and some h}, R(K) = {A : A ∼ = C and h : B ³s C for some B ∈ K and some h}, E(K) = {A : A ∼ = C and h : C ³s B for some B ∈ K and some h},

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Q P (K) = {A : A ∼ = i∈I Ai and Ai ∈ K for all i ∈ I}, Q Pf (K) = {A : A ∼ = i∈I Ai /F, Ai ∈ K for all i ∈ I and F is a proper filter on I}, Q Pu (K) = {A : A ∼ = i∈I Ai /U, Ai ∈ K for all i ∈ I and U is an ultrafilter on I}, Q Psd (K) = {A : h : A ½sd i∈I Ai for some h, Ai ∈ K for all i ∈ I}, Q Pf s (K) = {A : A ∼ Ai , Ai ∈ K for all i ∈ I, = B/F, B ⊆sd i∈I

F is a proper filter on I}. If O and O0 are any two of these operators, we write OO0 for their composition and O ≤ O0 to mean that O(K) ⊆ O0 (K) for any class K of L-structures. For each O, we also use the short notation O∗ for LO. Note that O(K), for O ∈ {P, Pf , Pu , Psd , Pf s }, are always nonempty classes, even if K is empty, since one can choose I = ∅ and then they contain the trivial, one-element structure where all relations are non-empty. When necessary, we shall write O(K) to indicate that we only take non-empty index sets in the respective constructions. Lemma 4.1. If O is any one of the operators defined above, O2 =O. Proof. The equality is easy to verify except when O is one of the operators that corresponds to a product construction. So let us give an idea of the proof for these cases. Assume first that O = Pf . Obviously, Pf ≤ Pf Pf . To prove the reverse inclusion, let Fj be a properSfilter on Ij , for all j ∈ J, and F a proper filter on J. Define a new index set K = j∈J (Ij × {j}) and let S G = { j∈J 0 (Fj × {j}) : J 0 ∈ F and Fj ∈ Fj for each j ∈ J 0 }. It is easy to see that G is a proper filter of Sb(K), so it suffices to show that Q (i,j)∈K

Q Q Aij /G ∼ = j∈J ( i∈Ij Aij /Fj )/F,

for all L-structures Aij . We shall give the preciseQ definition of the isomorphism and omit the details. Let a = haij : (i, j) ∈ Ki ∈ (i,j)∈K Aij . For each j ∈ J, let Q Q aj := a ¹ Ij × {j}. Clearly aj ∈ i∈Ij Aij , so that aj /Fj ∈ i∈Ij Aij /Fj . Define h to be the function given by h(a/G) := haj /Fj : j ∈ Ji/F. Then h is the desired isomorphism. The above construction specializes trivially to the case that the filters Fj and F are respectively {Ij } and {J}, so that we also have a proof of the equality P = P P . Moreover, if Fj , for j ∈ J, and F are all ultrafilters, the set G is again an ultrafilter of Sb(K), and hence the idempotency of PuQis also proved. Q Suppose now O = Psd . Let hj : Aj ½sd i∈Ij Aij and h : A ½sd j∈J Aj . If K S denotes again the set j∈J (Ij × {j}), we already know that Q j∈J

¡Q i∈Ij

Aij

¢

Q ∼ = (i,j)∈K Aij .

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So it is enough to verify that there exists a subdirect embedding from A into the Q Q product j∈J ( i∈Ij Aij ). Indeed, the mapping given by a 7−→ hhj πj ha : j ∈ Ji satisfies the desired condition. Finally, the idempotency of the operator Pf s can be derived almost immediately from the equalities already stated; we omit the details.  The second of the lemmas describes the behavior of the operators E and R when composed with some other operators. Lemma 4.2. For all O ∈ {S, Se , P, Pf , Pu , Psd }, the following is true. (i) OE ≤ EO; (ii) If O 6= Psd , then OR ≤ RO. Proof. (i) Suppose first A ∈ SE(K). Let h : C ³s B and A ⊆ C for some B ∈ K. By 1.2, hA ⊆ B. Moreover, the restriction of h to A defines a reductive homomorphism from A onto hA, as h−1 rhA = h−1 hrA by the definition of hA. Hence, A ∈ ES(K). This gives the statement for O = S. To show that Se E(K) ⊆ ESe (K), let A be such that A ⊆e C and h : C ³s B for some B ∈ K. The restriction of h to A is still an elementary homomorphism, and hence hA ⊆e B. Therefore, A ∈ ESe (K). Q : Ai ³s Bi Let us consider now the case O = P . Suppose A ∼ = Q i∈I Ai and hiQ with Bi ∈ K, for any i ∈ I. Define the mapping h from i∈I Ai intoQ i∈I Bi by ha = hhi ai : i ∈ Ii, where a = hai : i ∈ Ii is an arbitrary element of i∈I Ai . We already know that h defines a surjective algebra homomorphism, so let us see Q that h is strong. Take an n-ary relation symbol r ∈ R and elements a1 , . . . , an ∈ i∈I Ai . Then ha1 , . . . , an i ∈ h−1 r

Q i∈I

Bi

iff hhi a1i , . . . , hi ani i ∈ rBi for all i ∈ I.

Thus, since hi is strong for each i ∈ I, theQdefinition of the product structure implies i∈I Ai , and hence h is strong. As a result, that this is equivalent to ha1 , . . . , an i ∈ r Q Q there is a reductive homomorphism from i∈I Ai onto i∈I Bi , and consequently A ∈ EP (K). The above proof extends easily to the cases O = Pf , Pu . Now, given any proper filter an untrafilter) of Sb (I) we define the canonical mapping hF from Q F (possiblyQ A /F into i i∈I i∈I Bi /F by hF (a/F) Q= (ha)/F, andQwe can verify that hF is again a reductive homomorphism from i∈I Ai /F onto i∈I Bi /F. In particular, the fact that h is strong follows from the equality {i ∈ I : hhi a1i , . . . , hi ani i ∈ rBi } = {i ∈ I : ha1i , . . . , ani i ∈ rAi }. Finally,Qassume A ∈ Psd E(K). Let hi : Ai ³sQ Bi for Bi ∈ K and i ∈ I, and let g : A ½sd i∈I Ai . Define the map h from A into i∈I Bi by letting ha = hhi ◦g(a) : i ∈ Ii. So defined, h is the composition of two strong homomorphisms (remember that any subdirect embedding is strong), so it is strong. Hence, A/Ker h ∼ = hA ⊆ Q B . On the other hand, the commutativity of the diagram involved implies i i∈I Q that the composition of h with the projection from i∈I Bi onto Bi is surjective,

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for all i. Consequently, hA is a subdirect product of {Bi : i ∈ I} and A ∈ EPsd (K). This completes the proof of part (i). (ii) Let A ⊆ C and h : B ³s C for some B ∈ K. As h is strong, Lemma 1.2 says that A0 = h−1 A is a substructure of B and the restriction of h to A0 is a reductive homomorphism. So A belongs to RS(K). Assume now that A is in addition an elementary substructure of C and let us see that A ∈ RSe (K). We continue with the previous notation. Let ϕ = ϕ(x1 , . . . , xk ) be any formula over L and let a01 , . . . , a0k ∈ A0 . As h : A0 ³s A, we have that A0 |= ϕ(x1 , . . . , xk ) [a01 , . . . , a0k ] iff A |= ϕ(x1 , . . . , xk ) [ha01 , . . . , ha0k ], by 3.1.1. Similarly, B |= ϕ(x1 , . . . , xk ) [a01 , . . . , a0k ] is equivalent to C |= ϕ(x1 , . . . , xk ) [ha01 , . . . , ha0k ]. So, as A ⊆e C, we conclude that A0 ⊆e B and, consequently, A ∈ R(A0 ) ⊆ RSe (K). The proof of the inequalities OR ≤ RO, for O ∈ {P, Pf , Pu }, is again straighforward and it is omitted.  Notice that the inequality Psd R ≤ RPsd does not seem to hold in general because, if Bi is a reduction of Ai for i ∈ I, once Q we lift a subdirect product of the family {Bi : i ∈ I} up to a substructure of i∈I Ai , we cannot guarantee the restriction of the projection into Ai to be surjective. Algebraically, filter extensions (and hence homomorphic images) does not behave as well as the other operators. This will appear obvious below in the study of elementary classes axiomatized by atomic formulas. The next lemma contains some of the properties of filter extensions that we shall need in the investigation of these classes. Lemma 4.3. (i) EF ≤ F E. (ii) F R ≤ RF = H. (iii) F S ≤ SF . Proof. (i) Suppose A ∈ EF (K) and let h : A ³s C, where B 4 C for some B ∈ K. Then h−1 rB ⊆ rA for all r ∈ R, so that A is a filter extension of hA, h−1 RB i. Moreover, hA, h−1 RB i ∈ E(B). Hence A ∈ F E(K). (ii) Let A ∈ F R(K) and let h : B ³s C and C 4 A for some B ∈ K. The preimage of A by h is a filter extension of B and, consequently, A ∈ RF (K). The equality RF = H has already been proved in Section 3. (iii) Assume C 4 A and C ⊆ B for some B ∈ K (in fact, we should suppose A is isomorphic to some filter extension of C and C isomorphic to some substructure of B, but the same argument can be applied). Define D = hB, RB ∪ RA i. Then B 4 D and, as can be easily proved, A ⊆ D. Consequently A ∈ SF (K).  Lemma 4.4. (i) EL = ER = RE. (ii) LE = LR = RL = L ≤ EL. Proof. (i) Assume A ∈ RE(K) and let C be such that A, B ∈ R(C) for some B ∈ K. By 3.4, C∗ ∼ = A∗ and C∗ ∼ = B∗ . Thus A, B ∈ E(C∗ ) and hence A ∈ ER(K). For the converse, let h : A ³s C and g : B ³s C, with B ∈ K. From universal algebra we know that there exists an absolutely free algebra F and surjective homomorphisms k : F ³A and f : F ³B such that h ◦ k = g ◦ f . So it suffices to define F =

17

hF, (h ◦ k)−1 RC i; the condition A, B ∈ R(F) holds and, consequently, A ∈ RE(K). This proves the equality ER = RE. To show that ER = EL, assume as before that A, B are expansions of some C, for B ∈ K. Then A, B are also expansions of C∗ , so that A ∈ EL(K). The opposite inclusion is trivial. (ii) It is a direct consequence of the definitions involved.  4.2. The Diagram Lemma. For languages with equality any elementary homomorphism is an embedding, so that the definition of elementary homomorphism may be reformulated by saying that a map h : A → B is elementary if h is an isomorphism of A onto an elementary substructure of B. This is not true, however, if the language does not involve the equality symbol ≈. We are going to see what happens in this case. For this purpose, we must introduce some definitions. Let A be an L-structure, and let LA be an A-expansion of L, i.e., the language obtained from L by adding new distinct individual constants ca for all a ∈ A. Following a common notation, all over this subsection we use a to indicate the sequence of elements of A according to a certain well ordering on A, and c to indicate the corresponding sequence of constants. Structures over LA are denoted (B, ba )a∈A , where B is an structure over L and ba is a member of B for each a ∈ A. As usual, we call diagram of A, denoted DA, the set of all atomic sentences and negations of atomic sentences over LA which hold in (A, a)a∈A . We define the Leibniz diagram of A, and denote it by Dl A, as the set that results from DA by adding all LA -sentences of the form ψ(t, t0 ), for ψ(x, y) a Leibniz L-formula and t, t0 closed terms of LA (i.e., terms constructed only from constants and function symbols of LA ) such that their interpretations in (A, a)a∈A are congruent modulo ΩA. This can be expressed as follows: Dl A = DA ∪ {ψ(t, t0 ) : ψ(x, y) is a Leibniz L-formula, t = t(c) and t0 = t0 (c), and tA (a) ≡ t0A (a) (ΩA)}. Finally, we call elementary diagram of A, De A, the set of all sentences of LA which hold in (A, a)a∈A . Note that by 2.6, Dl A ⊆ De A. The following theorem shows that, whereas the nature of elementary diagrams does not depend on the presence of the equality symbol in the language, the weaker concept of diagram, as a logical expression of the notion of substructure when L has equality, needs to be replaced by that of Leibniz diagram if L has no equality. This fact is largely used to prove the main results in Section 5. Theorem 4.5. (Diagram Lemma) The following hold for all L-structures A, B. (i) If (B, ha)a∈A is a model of Dl A then h∗ : A∗ ½s B∗ . (ii) If (B, ha)a∈A is a model of De A then h∗ : A∗ ½e B∗ . Moreover, implications become equivalences under the assumption that h is a homomorphism from A into B. Proof. (i) Assume (B, ha)a∈A is a model of Dl A and a∗ = a0∗ , for some a, a0 ∈ A. By Theorem 2.6, ψ(ca , ca0 ) ∈ Dl A for all Leibniz L-formulas ψ(x, y). Thus

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B |= ψ(x, y) [ha, ha0 ] and consequently (ha)∗ = (ha0 )∗ . This proves that h∗ is well defined. Let us see that h∗ is a strong homomorphism. For this, let f be an n-ary function symbol and a = ha1 , . . . , an i a member of An . For every Leibniz formula ψ(x, y) we have ψ(cf A a1 ...an , f ca1 . . . can ) ∈ Dl A. Consequently, since (B, ha)a∈A is a model of Dl A, hf A a1 . . . an ≡ f B ha1 . . . han (ΩB), which implies that h∗ is a homomorphism between the underlying quotient algebras. Similarly, if r is an n-ary relation symbol, the condition rca1 . . . can ∈ Dl A iff (B, ha)a∈A |= rca1 . . . can follows directly from the definition of Dl A and the fact that (B, ha)a∈A is a model ∗ ∗ of Dl A. So a ∈ rA iff (ha)∗ ∈ rB , and h∗ is strong. Finally, Proposition 2.2 implies that Ker h∗ ∈ Co A∗ . Hence, Ker h∗ = ∆A∗ and h∗ is a strong embedding from A∗ into B∗ . The reverse implication is an easy consequence from the definitions involved. Given an n-ary relation symbol r and elements a1 , . . . , an ∈ A, the condition (A∗ , a∗ )a∈A |= rca1 . . . can is equivalent to (B∗ , (ha)∗ )a∈A |= rca1 . . . can , because h∗ is strong. Hence, (A, a)a∈A |= rca1 . . . can iff (B, ha)a∈A |= rca1 . . . can . On the other hand, let t and t0 be terms over LA whose constants are among ca1 , . . . , cak for some a1 , . . . , ak ∈ A. If ψ(t, t0 ) ∈ Dl A then tA (a1 , . . . , ak ) ≡ t0A (a1 , . . . , ak ) (ΩA). Since h is a homomorphism and h∗ an embedding, the last condition implies that tB (ha1 , . . . , hak ) ≡ t0B (ha1 , . . . , hak ) (ΩB), and hence, by Theorem 2.6, (B, ha)a∈A satisfies ψ(t, t0 ). Therefore, (B, ha)a∈A is a model of Dl A. This completes the proof of (i). (ii) The fact that h∗ is elementary follows from Proposition 1.3. According to this proposition, (A∗ , a∗ )a∈A ≡ (A, a)a∈A and (B∗ , (ha)∗ )a∈A ≡ (B, ha)a∈A , so for all L-formulas ϕ(x1 , . . . , xk ) and all a1 , . . . , ak ∈ A, we have (A∗ , a∗ )a∈A |= ϕ(ca1 , . . . , cak ) iff (B∗ , (ha)∗ )a∈A |= ϕ(ca1 , . . . , cak ). Thus, A∗ |= ϕ(x1 , . . . , xk ) [a∗1 , . . . , a∗k ] iff B∗ |= ϕ(x1 , . . . , xk ) [(ha1 )∗ , . . . , (hak )∗ ], and the only-if part is proved. The converse is obtained by a similar argument.



Corollary 4.6. Let A, B be L-structures. Then: (i) h : A →e B implies h : A →s B and hA ⊆e B; (ii) h : A →e B implies h∗ : A∗ ½e B∗ and h∗ A∗ ⊆e B∗ . Proof. Using Lemma 1.2, elementary, (B, ha)a∈A is embedding. Moreover, A∗

part (i) is easy to check. Let us show (ii). Since h is a model of De A. Hence, by 4.5(ii), h∗ is an elementary ∼ = h∗ A∗ and A∗ ≡ B∗ imply that h∗ A∗ ≡ B∗ . 

19

Observe that, according to the preceding corollary, if h : A →e B then some quotient of A is isomorphic to some elementary substructure of B, whereas the Leibniz quotient A∗ of A is directly isomorphic, as it occurs when the language has equality, to some elementary substructure of the Leibniz quotient B∗ of B. We shall see in the next section that much of the difference between the algebraic characterization of certain classes of structures and the characterization of the corresponding classes defined using the equality symbol has to do with this fact. 4.3. The Reduction Operator Lemma. The last fundamental lemma concerns the special commutativity properties of the reduction operator L when composed with other operators. This sort of commutativity is essential in order to derive in the following section the Birkhoff-type characterizations of reduced model classes from the corresponding characterizations for full classes. Theorem 4.7. (Reduction Operator Lemma) (i) For each operator O ∈ {S, P, Pf , Pu , Psd }, we have LO = LOL, i.e., O∗ = O∗ L. (ii) LSe = LSe L = Se L. Proof. (i) Assume first O = S and let A ∈ S ∗ (K). Suppose A ∼ = C∗ for some C such that C ⊆ B and B ∈ K. We need a sublemma whose proof is immediate: If A ⊆ B and θ ∈ Co B, then the mapping a/θA 7−→ a/θ defines a strong embedding from A/θA into B/θ, where θA = θ ∩ A2 . In our case, the sublemma says that there is a strong embedding from C/θC into B∗ , where θC = ΩB ∩ C 2 . So, by virtue of 1.2 and the Homomorphism Theorem stated in 3.3, C/θC is isomorphic to some substructure of B∗ . On the other hand, Proposition 3.4 implies C∗ ∼ = (C/θC )∗ and consequently A ∼ = (C/θC )∗ . ∗ ∗ ∼ Thus, A ∈ S L(K). To prove the reverse inclusion, let A = C for some C such that C ⊆ B∗ and B ∈ K. If πB denotes the projection from B onto B∗ then −1 C0 := πB C is a substructure of B and the restriction of πB to C0 is also a reductive homomorphism. Therefore we have A ∈ S ∗ (K). Let us suppose now that O = Pf . We shall prove that for each family of Lstructures {Ai : i ∈ I} and each proper filter F over I, we have (4)

Q Q ( i∈I Ai /F)∗ ∼ = ( i∈I A∗i /F)∗ .

Under this assumption, the desired equality follows trivially, for A ∈ Pf∗ (K) iff Q Q A∼ = ( i∈I A∗i /F)∗ = ( i∈I Ai /F)∗ for some Ai ∈ K, i ∈ I, and A ∈ Pf∗ L(K) iff A ∼ for some Ai ∈ K, i ∈ I. So let us proceed to prove (4). b and A respectively the products Q A∗ and Q Ai , and define a Denote by A i∈I i i∈I b b = ha∗i : mapping h from A/F into (A/F)∗ by h(b a/F) = (a/F)∗ , for every element a b b/F = i ∈ Ii ∈ A. We must first of all show that h is well defined. For this, assume a b b/F, i.e., {i ∈ I : a∗i = b∗i } ∈ F , and let us conclude that (a/F)∗ = (b/F)∗ . We use Theorem 2.6. Given any atomic L-formula ϕ := ϕ(x, z1 , . . . , zk ) and elements

20

a1 /F, . . . , ak /F ∈ A/F, Theorem 1.4 says that A/F |= ϕ(x,z1 , . . . , zk ) [a/F, a1 /F, . . . , ak /F] iff {i ∈ I : Ai |= ϕ(x, z1 , . . . , zk ) [ai , a1i , . . . , aki ]} ∈ F. On the other hand, {i ∈ I : a∗i = b∗i } ∩ {i ∈ I : Ai |= ϕ(x, z1 , . . . , zk ) [ai , a1i , . . . , aki ]} ⊆ {i ∈ I : Ai |= ϕ(x, z1 , . . . , zk ) [bi , a1i , . . . , aki ]}. Therefore, since F is a filter, A/F |= ϕ(x, z1 , . . . , zk ) [a/F, a1 /F, . . . , ak /F] and b b/F = b/F a implies that {i ∈ I : Ai |= ϕ(x, z1 , . . . , zk ) [bi , a1i , . . . , aki ]} ∈ F, which is the same as A/F |= ϕ(x, z1 , . . . , zk ) [b/F, a1 /F, . . . , ak /F], again by 1.4. b b/F = b/F Consequently, under the assumption a we conclude that A/F |= ϕ(x, z1 , . . . , zk ) →ϕ(y, z1 , . . . , zk ) [a/F, b/F, a1 /F, . . . , ak /F]. The same argument proves the reverse implication. So, as ϕ(x, z1 , . . . , zk ) and a1 /F, . . . , ak /F are arbitrary, Theorem 2.6 gives (a/F)∗ = (b/F)∗ . The fact that h is a strong homomorphism is a direct consequence of the definitions involved and the proof is omitted. Finally, since h is surjective, Proposition 3.4 says that ∗ ∼ b h∗ : (A/F) = (A/F)∗

and hence h∗ is the desired isomorphism. This completes the proof of (4). The equalities O∗ = O∗ L for O ∈ {P, Pu } can also be proved Q by the same argument. So consider finally the case O = Psd . Let g : A ½sd i∈I Ai with ∗ ∗ ∗ ∗ Ai ∈ K, for i ∈ I, so that A∗ ∈ Psd (K). We Q are ∗going to show that A ∈ Psd (K ). Indeed, consider the map h from A into i∈I Ai defined as follows: if a ∈ A and ga = hai : i ∈ Ii, let ha = ha∗i : i ∈ Ii. Clearly h is a strong homomorphism and its composition withQ the projection from Ai into A∗i is surjective, for all i. Therefore A/Ker h ½sd i∈I A∗i , and hence ¡ ¢∗ ∗ A/Ker h ∈ Psd (K∗ ). Proposition 3.4 completes the proof. (ii) For O = Se we reason in very much the same manner as for O = S and then apply 1.3 to obtain Se∗ = Se∗ L. To be more precise, let us keep the same notation and assume C ⊆e B. Then, since C/θ ≡ C and B ≡ B∗ , we have C/θ ≡ B∗ and consequently the embedding from C/θ into B∗ is elementary. For the converse we just need to check that if C ≡ B∗ , the preimage of C by πB is also elementarily equivalent to B. And this is a straightforward verification Once we have derived the equality Se∗ = Se∗ L, it is easy to see that Se∗ L = Se L. Indeed, let us prove that if A ⊆e B and B is reduced then A is reduced. Two applications of 2.6 give the following: for all a, b ∈ A, a ≡ b (ΩA) iff A |= ψ(x, y) [a, b] for each Leibniz L-formula ψ iff B |= ψ(x, y) [a, b] for each Leibniz L-formula ψ iff a ≡ b (ΩB).

21

So ΩA ⊆ ΩB, and consequently if B is reduced then A is reduced as well.



5. The Main Theorems In Section 3 it has been established that first-order logic without equality has two complete semantics, viz., the full and the reduced semantics. This suggests the problem of finding what the difference is, if any, between the whole class of models and the class of reduced models of a given set of sentences, see [10] and [11]. A part of this problem is solved in this section. We state characterizations, in the style of Birkhoff’s Variety Theorem (see, e.g., [3, Thm. II.11.9]), of both the full and the reduced model classes of certain theories; namely, elementary, universal, universal Horn and universal atomic theories. To this end, we first prove algebraic characterizations for the full classes and then, using the Reduction Operator Lemma, we derive analoguous results for the reduced classes. 5.1. Elementary Classes. Remember that a class K of L-structures is said to be elementary if there exists some set Γ of sentences over L such that K= M od Γ, or equivalently, if K = M od T h K. Thus, the following theorem is an extension to general first-order languages, with or without equality, of a well known result in classical model theory. Theorem 5.1. The following are equivalent for any class K of L-structures. (i) K is an elementary class. (ii) K is closed under E, R, Se and P u . (iii) K = ERSe P u (K0 ), for some class K0 . Proof. The implication from (i) to (ii) follows directly from Proposition 1.3 and L Ã os Theorem on ultraproducts. Moreover, (ii) implies (iii) is trivial by taking K0 = K. So let us show that (iii) entails (i). We claim that K is axiomatizable by T h K0 , where K0 is as in (iii). Note first of all that for any class L of L-structures, T h L = T h O(L) whenever O ∈ {E, R, Se , P u }, again by 1.3 and 1.5. Thus, T h K = T h K0 and the inclusion K ⊆ M od T h K0 is clear. Assume A ∈ M od T h K0 and let us see that A ∈ K. Let ∆ = Sbω (De A). Given any set Φ of LA -formulas, we write Φ(ca1 , . . . , cak ) to mean that the constants ca , for a ∈ A, appearing in the elements of Φ are among ca1 , . . . , cak . We claim that if Φ ∈ ∆, then there exist some BΦ ∈ K0 and some {ba,Φ : a ∈ A} ⊆ BΦ such that V (BΦ , ba,Φ )a∈A |= Φ(ca1 , . . . , cak ). Suppose not. Then, given any B ∈ K0 and any {ba : a ∈ A}, we have V (B, ba )a∈A |= ¬ Φ(ca1 , . . . , cak ). V Consequently, the class K0 satisfies the L-sentence ∀x1 . . . ∀xk ¬ Φ(x1 , . . . , xk ), i.e., V ∀x1 . . . ∀xk ¬ Φ(x1 , . . . , xk ) ∈ T h K0 .

22

V But this implies that A |= ∀x1 . . . ∀xk ¬ Φ(x1 , . . . , xk ), and hence contradicts the assumption Φ ∈ ∆. So the claim does hold. As usual, define JΦ = {Ψ ∈ ∆ : Φ ⊆ Ψ} for Φ ∈ ∆. The family {JΦ : Φ ∈ ∆} has the finite intersection property, so there is an ultrafilter U on ∆ such that JΦ ∈ U Q for every Φ. Let B =Q Φ∈∆ BΦ /U. Clearly B ∈ P u (K0 ). Let us show that if ba := hba,Φ : Φ ∈ ∆i ∈ Φ∈∆ BΦ , for each a ∈ A, then (5)

(B, ba /U )a∈A is a model of De A.

Indeed, suppose ϕ := ϕ(ca1 , . . . , cak ) ∈ De A. The following equivalences hold (the second one by L Ã os Theorem): (B, ba /U)a∈A |= ϕ(ca1 , . . . , cak ) iff B |= ϕ(x1 , . . . , xk ) [ba1 /U, . . . , bak /U] iff {Φ ∈ ∆ : BΦ |= ϕ(x1 , . . . , xk ) [ba1 ,Φ , . . . , bak ,Φ ]} ∈ U iff {Φ ∈ ∆ : (BΦ , ba,Φ )a∈A |= ϕ(ca1 , . . . , cak )} ∈ U . Also, J{ϕ} ∈ U and J{ϕ} ⊆ {Φ ∈ ∆ : (BΦ , ba,Φ )a∈A |= ϕ(ca1 , . . . , cak )}. Therefore, since U is an ultrafilter, the last condition above is satisfied. So (B, ba /U)a∈A is a model of ϕ(ca1 , . . . , cak ) and (5) is proved. We now apply the Diagram Lemma. Then h∗ : A∗ ½e B∗ , where ha = (ba /U), and so 4.6 gives A ∈ ESe R P u (K0 ). Lemma 4.2(ii) and the assumption that K= ERSe P u (K0 ) complete the proof.  Given a class K of L-structures, we define the full elementary class generated by K, or simply the elementary class generated by K, as ¡KE = ¢∗ M od T h K. Also, we call reduced elementary class generated by K the class KE = L(M od T h K); ¡ ¢∗ observe that KE is not in general elementary. The next corollary describes the ¡ ¢∗ way to contruct KE and KE from K by applying certain operators. Corollary 5.2. The following holds for any class K of L-structures. (i) KE = ERSe P u (K). ¡ ¢∗ ∗ (ii) KE = Se P u (K∗ ). Proof. Part (i) follows immediately from the proof of the preceding theorem, because it establishes the equality M od T h K = ERSe P u (K). To see (ii), it suffices to show ∗ that LERSe P u = Se P u L. Indeed, LERSe P u = LSe P u , =

∗ Se P u L,

by Lemma 4.4, by Theorem 4.7.



Corollary A class K of reduced L-structures is a reduced elementary class ¡ ¢5.3. ∗ (i.e., K= KE ) iff it is closed under elementary substructures and reduced ultraproducts modulo ultrafilters over nonempty sets. Proof. It is an obvious consequence of Corollary 5.2(ii).



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A reduced elementary class is not in general closed under the operator P u . A counterexample is provided by Blok and Pigozzi [2, p. 30]; actually, they give a universal Horn theory and describe an ultraproduct of reduced models which is not reduced. 5.2. Universal Classes. Recall that a class K of L-structures is said to be universal if there exists some set Γ of universal L-sentences such that K = M od Γ, or equivalently, if K = M od T h∀ K, where T h∀ K denotes the whole set of universal L-sentences satisfied by all members of K. The following is the characterization of universal classes defined with or without equality; it simultaneously extends a well known result in classical model theory (see, e.g., [3, Thm. V.2.16]) and a more recent result of Czelakowski [5, Thm. I.7]. Theorem 5.4. For any class K of L-structures, the following are equivalent. (i) K is a universal class. (ii) K is closed under E, R, S and P u . (iii) K = ERS P u (K0 ), for some class K0 . Proof. The implication from (i) to (ii) follows from 1.3, L Ã os Theorem and the additional well-known fact that universal sentences are preserved under substructures. (ii) implies (iii) is again trivial. So let us concentrate on the proof that (iii) entails (i). We follow a similar argument to the one given for Theorem 5.1. In this case, the aim is to see that K is axiomatizable by T h∀ K0 . Note again that the inclusion K ⊆ M od T h∀ K0 holds. Thus assume A ∈ M od T h∀ K0 and let us show A ∈ K. Let ∆ = Sbω (D A). For every Φ ∈ ∆, there exist some BΦ ∈ K0 and some {ba,Φ : a ∈ A} ⊆ BΦ such that V

(BΦ , ba,Φ )a∈A |=

Φ(ca1 , . . . , cak );

V otherwise, we could conclude that ∀xV Φ(x1 , . . . , xk ) ∈ T h∀ K0 , which is 1 . . . ∀xk ¬ impossible because A |= ∃x1 . . . ∃xk Φ(x1 , . . . , xk ). Define as before JΦ = {Ψ ∈ ∆ : Φ ⊆ Ψ} for every Φ ∈ ∆, and let U be an ultrafilter on ∆ containing the family {JΦ : Φ ∈ ∆}. Let B :=

Q Φ∈∆

BΦ /U,

ba := (ba,Φ : Φ ∈ ∆) ∈

Q Φ∈∆

BΦ ,

for each a ∈ A,

C := B ¹ {ba /U : a ∈ A}. Clearly C ∈ S P u (K0 ). Let us establish the following lemma: (6)

(C, ba /U )a∈A is a model of Dl A.

We begin by showing that (C, ba /U )a∈A is a model of DA. Consider any element ϕ := ϕ(ca1 , . . . , cak ) of DA. We have that J{ϕ} ∈ U and J{ϕ} ⊆ {Φ ∈ ∆ : BΦ |= ϕ(x1 , . . . , xk ) [ba1 ,Φ , . . . , bak ,Φ ]}.

24

Hence, since U is an ultrafilter, the last set belongs to U . So, by virtue of 1.5, B |= ϕ(x1 , . . . , xk ) [ba1 /U, . . . , bak /U ] and consequently, since ϕ is an atomic or negated atomic L-formula, C |= ϕ(x1 , . . . , xk ) [ba1 /U, . . . , bak /U]. Finally, this last condition is equivalent to (C, ba /U )a∈A |= ϕ(ca1 , . . . , cak ). Now consider any other element ψ(t, t0 ) of Dl A, where t := t(ca1 , . . . , cak ) and 0 t := t0 (ca1 , . . . , cak ) for some k > 0 and some a1 , . . . , ak ∈ A. Our definition of Leibniz diagram says that we have tA (a1 , . . . , ak ) ≡ t0A (a1 , . . . , ak ) (ΩA). Assume ψ(x, y) := ∀z1 . . . ∀zp (ϕ(x, z1 , . . . , zp ) ↔ ϕ(y, z1 , . . . , zp )) and take arbitrary elements b1 /U, . . . , bp /U of C. We must prove the equivalence (C, ba /U)a∈A |= ϕ(t(ca1 , . . . , cak ), z1 , . . . , zp ) [b1 /U , . . . , bp /U] (7)

iff (C, ba /U)a∈A |= ϕ(t0 (ca1 , . . . , cak ), z1 , . . . , zp ) [b1 /U, . . . , bp /U].

Since C is generated by {ba /U : a ∈ A}, there exist some q ≥ 0, some a01 , . . . , a0q ∈ A and some L-terms t1 , . . . , tp in q variables such that bi /U := tC i (ba01 /U, . . . , ba0q /U),

for 1 ≤ i ≤ p.

Thus we have the following chain of equivalences: (C, ba /U)a∈A |= ϕ(t(ca1 , . . . , cak ), z1 , . . . , zp ) [b1 /U, . . . , bp /U ] iff C |= ϕ(t(x1 , . . . , xk ), z1 , . . . , zp ) [ba1 /U, . . . , bak /U , b1 /U, . . . , bp /U] iff C |= ϕ(t(x1 , . . . , xk ), t1 (u1 , . . . , uq ), . . . , tp (u1 , . . . , uq )) [ba1 /U, . . . , bak /U, ba01 /U, . . . , ba0q /U] (u1 , . . . , uq are additional variables distinct from x1 , . . . , xk ). Take y to be some other new variable and let σ be the atomic L-formula given by σ(y, u1 , . . . , uq ) := ϕ(y, t1 (u1 , . . . , uq ), . . . , tp (u1 , . . . , uq )). Then the last condition above can be expressed as C |= σ(t(x1 , . . . , xk ), u1 , . . . , uq ) [ba1 /U, . . . , bak /U , ba01 /U, . . . , ba0q /U]. Hence, since it has already been proved that (C, ba /U)a∈A is a model of DA, we have A |= σ(t(x1 , . . . , xk ), u1 , . . . , uq ) [a1 , . . . , ak , a01 , . . . , a0q ], i.e., A |= σ(x, u1 , . . . , uq ) [tA (a1 , . . . , ak ), a01 , . . . , a0q ].

25

We now apply the assumption tA (a1 , . . . , ak ) ≡ t0A (a1 , . . . , ak ) (ΩA), which says that the preceding condition is equivalent to (8)

A |= σ(x, u1 , . . . , uq ) [t0A (a1 , . . . , ak ), a01 , . . . , a0q ],

Finally, reversing the previous argument, we derive the equivalence of (8) with the right-hand side of (7): (C, ba /U )a∈A |= ϕ(t0 (ca1 , . . . , cak ), z1 , . . . , zp ) [b1 /U, . . . , bp /U]. This completes the proof of (6). Apply now part (i) of Diagram Lemma to (6). We have that the mapping a∗ 7−→ (ba /U)∗ defines a strong embedding from A∗ into C∗ . Moreover, h is surjective, so that once more the Homomorphism Theorem gives A∗ ∼ = C∗ . As a result, A ∈ 0 ERS P u (K ) = K and the theorem is proved.  Let K be any class of L-structures. The full universal class generated by K, or simply the universal class generated by K, is defined as KU¡ = M ¢∗ od T h∀ K, whereas the reduced universal class generated by K is taken to be KU = L(M od T h∀ K). ¡ ¢∗ Once more, KU need not be even an elementary class. The next result looks like Corollary 5.2. Corollary 5.5. If K is any class of L-structures, the following holds. (i) KU = ERSP u (K). ¡ ¢∗ ∗ (ii) KU = S ∗ P u (K∗ ). Proof. We just repeat the argument for the proof of Corollary 5.2.



Corollary ¡ ¢∗ 5.6. A class K of reduced L-structures is a reduced universal class (i.e., K = KU ) iff it is closed under reduced substructures and reduced ultraproducts modulo ultrafilters over nonempty sets.  A reduced universal class is not in general closed under the operators S and P u . For ultraproducts this is shown by the example given in the previous subsection. For substructures we can find simple counterexamples. For instance, consider the language of groups together with a unary relation symbol, L = { ·, e, r }. The whole class of L-structures is universal (the sentence ∀x(rx →rx) provides an axiomatization). Let A be a simple group and B a nonsimple subgroup of A. Then, if N is the universe of a normal subgroup of B, A = hA, N i is a reduced structure, B = hB, N i ⊆ A and B is not reduced. 5.3. Universal Horn Classes. Following the common terminology, let us call (strict) universal Horn sentence over L any universal L-sentence in prennex form whose matrix is the disjunction of a finite set (possibly empty) of negated atomic formulas and exactly one atomic formula. Then, we say that a class K of L-structures is a (strict) universal Horn class if there exists some set Γ of universal Horn Lsentences such that K= M od Γ, or equivalently, if K = M od T h∀H K, where T h∀H means the whole set of universal Horn sentences satisfied by the members of K.

26

Our purpose now is to provide some algebraic characterizations of universal Horn classes that hold for languages with as well as without equality. The results we are going to establish generalize the classical theorem of Mal’cev [15, p. 421] and some more recent theorems due to Czelakowski [6, 7]. The technique of the proof given here substantially differs from the one used by the second author. Theorem 5.7. The following are equivalent for any class K of L-structures. (i) K is a universal Horn class. (ii) K is closed under E, R, S and Pf . (iii) K= ERSPf (K0 ), for some class K0 . Proof. As for universal classes, (i) implies (ii) is easily checked using Theorem 1.4 instead of L Ã os Theorem. Likewise, (ii) implies (iii) is clear. Let us prove the implication from (iii) to (i). For this, we shall see that (iii) entails K is axiomatizable by T h∀H K0 . Certainly K ⊆ M od T h∀H K0 (observe that K must contain the trivial, one-element structure). Suppose A ∈ M od T h∀H K0 . Let ∆ = Sbω (DA). If we are given Φ ∈ ∆, Φ := Φ(ca1 , . . . , cak ), then A |= ∃x1 . . . ∃xk

V

Φ(x1 , . . . , xk ).

We want to show that some member of P (K0 ) satisfies this sentence as well. For this purpose it suffices to prove that (9)

∀x1 . . . ∀xk ¬

V

Φ(x1 , . . . , xk ) ∈ / T h P (K0 ).

We distinguish three cases. If none of the elements of Φ is a negated atomic Lformula then (9) holds, for P (K0 ) contains the trivial, one-element structure which does not satisfy the negation of any atomic L-formula. If exactly one element of Φ is negated atomic then the universal sentence above is logically equivalent to a universal Horn L-sentence which is not true in A and, consequently, since A ∈ M od T h∀H K0 , in K0 . The last case is the most difficult to argue. Let Φ := {ϕ1 , . . . , ϕq } and let us suppose that at least two elements of Φ are negated atomic formulas, say ϕi for 1 ≤ i ≤ p, where 2 ≤ p ≤ q. Then one can reason as above that ∀x1 . . . ∀xk (¬ϕi (x1 , . . . , xk ) ∨ ¬ϕp+1 (x1 , . . . , xk ) ∨ · · · ∨ ¬ϕq (x1 , . . . , xk )) ∈ / T h K0 , for 1 ≤ i ≤ p. Consequently, for some Bi ∈ K0 and some bi1 , . . . , bik ∈ Bi , 1 ≤ i ≤ p, Bi |= ϕi (x1 , . . . , xk ) ⊂ ϕp+1 (x1 , . . . , xk ) ⊂ . . . ⊂ ϕq (x1 , . . . , xk )) [bi1 , . . . , bik ]. Define bj := (b1j , . . . , bpj ) ∈

Q 1≤i≤p

Bi ,

1 ≤ j ≤ k.

Then Theorem 1.4 implies Q 1≤i≤p

and hence, since

Q 1≤i≤p

Bi |=

V

Φ(x1 , . . . , xk ) [b1 , . . . , bk ],

Bi ∈ P (K0 ), (9) is proved.

27

Now, for each Φ ∈ ∆, consider BΦ ∈ P (K0 ) and {ba,Φ : a ∈ A} ⊆ BΦ such that (BΦ , ba,Φ )a∈A |=

V

Φ(ca1 , . . . , cak ).

We can now proceed as in the proof of 5.4 to obtain an LA -structure (C, ba /U)a∈A of SPu P (K0 ) such that (C, ba /U)a∈A ∈ M od Dl A. So, a new application of the Diagram Lemma gives A ∈ ERSPu P (K0 ). But SPu P ≤ SPf Pf , = SPf ,

by definition, by Lemma 4.1.

Hence, A ∈ ERSPu P (K0 ) ⊆ ERSPf (K0 ) and the assumption (iii) says that A ∈ K. This completes the proof of the theorem.  Given any class K of L-structures, we define the full universal Horn class generQ ated by K, or simply the universal Horn class generated ¡by K, ¢ as K =∗M od T h∀H K, Q ∗ and the reduced universal Horn class generated by K as K = M od T h∀H K. The next result includes a generalization of [2, Thm. 6.2]. Corollary 5.8. The following is true for any class K of L-structures. (i) KQ = ERSPf (K). ¡ ¢∗ (ii) KQ = S ∗ Pf∗ (K∗ ).  Corollary 5.9. ¡ A ¢∗ class K of reduced L-structures is a reduced universal Horn class (i.e., K = KQ ) iff it is closed under both reduced substructures and reduced filtered products.  We use the notation Q to abbreviate the composed operator ERSPf , so we have just proved that KQ = Q(K) and Q∗ = S ∗ Pf∗ . The next lemmas can be used to derive some other useful descriptions of these operators Q and Q∗ for generating universal Horn classes. Lemma 5.10. (Gr¨atzer and Lasker [14, Lemma 2]) SPf = SP Pu .  Lemma 5.11. (Czelakowski [7]) SPf = Pf s = Psd SPu . Proof. Let us prove first the equality SPf = Pf s . The inclusion Pf s ≤ SPf is obvious: by definition, a filtered subdirect product of a system of structures is always isomorphic to a substructure of a filtered product of the system. Also, Pf ≤ Pf s . So let us see that S ≤ Pf s . Take an arbitrary class K of L-structures, and suppose A ⊆ B ∈ K. Define (10)

C := {b ∈ B ω : bi = a if i ≥ m, for some a ∈ A and m ∈ ω}.

Note that, for every b ∈ C, the element a in (10) is unique; let us denote it by a(b). Also, C is the universe of a subalgebra of the direct power Bω ; rather, it is the

28

universe of a subdirect power of Bω , for the projection of C into each component is surjective. So let C := Bω ¹ C and take F := {X ∈ Sb(ω) : X is finite }. We claim that the mapping h from C/F into A given by b/F 7−→ a(b) defines an isomorphism between the filtered subdirect power C/F and the substructure A. Indeed, if b, b0 ∈ C then b/F = b0 /F iff there exists m ∈ ω such that bi = b0i for all i ≥ m iff a(b) = a(b0 ). Thus h is well defined and bijective. Now choose elements b1 , . . . , bn ∈ C and let f and r be a function and a relation symbol, repectively, of arity n. Since a(f C b1 . . . bn ) = f B b1m . . . bnm for some m ∈ ω, we have that h(bi /F) = a(b) = bjm for all 1 ≤ j ≤ n, and consequently hf C/F b1 /F . . . bn /F = a(f C b1 . . . bn ) = f A a(b1 /F) . . . a(bn /F). Moreover, by the definition of filtered subdirect product, hb1 /F, . . . ,bn /Fi ∈ rC/F iff {i ∈ ω : hb1i , . . . , bni i ∈ rB } ∈ F iff there exists m ∈ ω such that hb1i , . . . , bni i ∈ rB for all i ≥ m iff ha(b1 ), . . . , a(bn )i ∈ rA . So h is the desired isomorphism. From the claim we conclude that A ∈ Pf s (B) and hence S(K) ⊆ Pf s (K). This completes the proof of the equality SPf = Pf s . To see Pf s = Psd SPu , we first notice that Psd SPu ≤ Pf s , for Pf s is idempotent by Lemma 4.1 and each one of the operators Psd , S and Pu is less than Pf s . For the reverse inclusion, let {Ai : i ∈ I} be a system of structures and let F be a proper filter on I. Clearly F may be expressed as the intersection of some family of ultrafilters on I; for simplicity suppose {Uj : j ∈ J} is such a family, T i.e., F = j∈J Uj , where Uj is an ultrafilter of Sb(I). Then the congruence ΘF QF is the intersection of the family {ΘUj : j ∈ J} and, for all r ∈ R, i∈I rAi = Q T QUj Ai r . So the filtered product i∈I Ai /F is subdirectly embeddable into ¢ Qj∈J ¡Qi∈I j∈J i∈I Ai /Uj by the mapping h : a/F 7−→ ha/Uj : j ∈ Ji. Thus, if A/F is a filtered subdirect product of the system {Ai : i ∈ I}, the image h(A/F) can be easily proved to be isomorphic to a subdirect product of the structures A/Uj , j ∈ J. In conclusion, Pf s ≤ Psd SPu , and this completes the proof of the second equality.  Corollary 5.12. The following equalities hold. (i) Q = ERSP Pu = ERPf s = ERPsd SPu .

29 ∗ ∗ ∗ (ii) Q∗ = S ∗ P ∗ Pu∗ = Pf∗s = Psd S Pu .

Proof. Part (i) follows directly from 5.10 and 5.11. To obtain (ii) we must apply Proposition 4.7(i) and the preceding lemmas.  The examples provided in the previous sections show that reduced universal Horn classes are not in general closed under the operators S and Pu . An easy counterexample borrowed from [2] proves that they are also not closed under P (and hence Pf ). Namely, if L consists of one relation symbol, of arity 1, and no function symbol, then the reduced L-structures are of the form A = h{a, b}, {a}i for distinct elements a, b. So A2 is not reduced, because |A2 | = 4. 6.4. Universal Atomic Classes. Let K be any class of L-structures. We say K is a universal atomic class if K = M od Γ for some set Γ of atomic formulas over L; equivalently, if K = M od Atm K, where Atm K denotes the whole set of atomic formulas satisfied by the members of K. The next result provides a generalization of Birkhoff’s Variety Theorem to general first-order languages, with or without equality. The proof is entirely of the same nature as the proof of the previous theorems in the section, and so is quite different from Birkhoff’s original proof. Theorem 5.13. For any class K of L-structures, the following are equivalent. (i) K is a universal atomic class. (ii) K is closed under H, E, S and P . (iii) K= HESP (K0 ), for some class K0 . Proof. (i) implies (ii) and (ii) implies (iii) are clear. Let us show (iii) implies (i) by proving that K is axiomatizable by Atm K0 . Once more the inclusion K ⊆ M od Atm K0 is easy to check. Assume A ∈ M od Atm K0 and let ∆ be the set D− A of negated atomic LA -sentences which are satisfied by (A, a)a∈A . Let ϕ := ϕ(ca1 , . . . , cak ) ∈ ∆. We claim there exist Bϕ ∈ K0 and {ba,ϕ : a ∈ A} ⊆ Bϕ such that (Bϕ , ba,ϕ )a∈A |= ϕ(ca1 , . . . , cak ). Otherwise, the sentence ∀x1 . . . ∀xk ¬ϕ(x1 , . . . , xk ) is logically equivalent to the universal closure of some member of Atm K0 , and hence A |= ∀x1 . . . ∀xk ¬ϕ(x1 , . . . , xk ). But this contradicts the assumption ϕ ∈ ∆. So let B :=

Q ϕ∈∆

Bϕ ,

ba := (ba,ϕ : ϕ ∈ ∆) ∈

Q ϕ∈∆

Bϕ , for each a ∈ A,

C := B ¹ {ba : a ∈ A}. Obviously C ∈ SP (K0 ). Moreover, by 1.4 we have B |= ∆ and thus C |= ∆. Consider the absolutely free L-algebra TeL,|A| over |A| variables {xa : a ∈ A}, and define h : T eL,|A| →C by xa 7−→ ba . Let F = hT eL,|A| , h−1 RC i be the preimage of C by h, so that we have h : F ³s C. We want the mapping xa 7−→ a to be a surjective homomorphism from F onto A. Clearly h is an algebra homomorphism.

30

Also, since (C, ba )a∈A is a model of ∆ = D− A, the following is true for any atomic L-formula rt1 . . . tn , where t1 , . . . , tn are terms in k variables: ht1 (xa1 , . . . , xak ), . . . , tn (xa1 , . . . , xak )i ∈ rF C C iff htC 1 (ba1 , . . . , bak ), . . . , tn (ba1 , . . . , bak )i ∈ r A A implies htA 1 (a1 , . . . , ak ), . . . , tn (a1 , . . . , ak )i ∈ r .

iff hht1 (xa1 , . . . , xak ), . . . , htn (xa1 , . . . , xak )i ∈ rA . Therefore, h : F ³A. Now we have that A ∈ H(F) and F ∈ E(C). Consequently, A ∈ HESP (K0 ).  Remark. Notice that this result specializes to Birkhoff’s Variety Theorem, for reductive homomorphisms are just isomorphisms when L has equality. In fact, the preceding proof simplifies in this case and provides a proof of a general form of Birkhoff’s Variety Theorem strictly based on model-theoretic techniques. The simplification goes as follows. If ≈ is a symbol of L, then the LA -sentence ¬ca ≈ ca0 belongs to ∆, for each a, a0 ∈ A such that a 6= a0 ; therefore, ba 6= ba0 must hold. Also, if ha1 , . . . , an i, ha01 , . . . , a0n i ∈ An , then f A a1 . . . an 6= f A a01 . . . a0n implies ¬f ca1 . . . can ≈ f ca01 . . . ca0n ∈ ∆, and hence f C ba1 . . . ban 6= f C ba01 . . . ba0n . In general, we can iterate this argument and prove that we can construct directly a surjective homomorphism from C onto A such that ba 7−→ a, and hence we obtain A ∈ H(C) ⊆ HSP (K0 ). We define the full universal atomic class generated by a class K, or simply the uniV versal atomic class generated by¡K, as ¢ K = M∗ od Atm K, and the reduced universal V ∗ atomic class generated by K as K = M od Atm K. Then we have: Corollary 5.14. The following is true for any class K of L-structures. V (i) K (K). ¡ V= ¢∗HESP (ii) K = F ∗ ESP (K). Proof. It follows directly from Lemmas 4.3(ii) and 4.4(ii).



As for universal Horn classes, we introduce the notation V to express the composed operator HESP , so that we have proved the equality KV = V (K). In general, however, the operators E and F do not commute, nor does F ∗ coincides with F ∗ L as occurs for the remaining operators (cf. the Reduction Operator Lemma above). There are easy counterexamples of this. For instance, let A := hN × N, +, (0, 0), ∼i,

B := hN × N, +, (0, 0), ∼0 i,

where ∼ is the binary relation on N×N given by (a, b) ∼ (a0 , b0 ) iff a + b0 = a0 + b, and ∼0 is the relation that results from ∼ by joining the set {(0, 1), (1, 0)}2 . Then it is easy to check that the Leibniz congruence on A is the relation ∼ (recall the

31

construction of integers by the symmetrization process). Also, ΩB coincides with the set of all pairs h(a, b), (a0 , b0 )i of ∼ that satisfy the following additional condition: (a, b) ∈ {(0, 0), (1, 0), (0,1)} or (a0 , b0 ) ∈ {(0, 0), (1, 0), (0, 1)} (11)

implies (a, b) = (a0 , b0 ).

Indeed, denote by θ such a set of pairs. Clearly θ is an equivalence relation and θ is included in ∼. So we have that θ is compatible with ∼0 . It remains to be shown that θ is also compatible with the addition. For this purpose, assume (a, b)θ(a0 , b0 ) and (c, d)θ(c0 , d0 ). We distinguish three cases. If none of the pairs (a, b), (c, d) belongs to {(0, 0), (1, 0), (0, 1)}, then we actually have that (a, b) ∼ (a0 , b0 ) and (c, d) ∼ (c0 , d0 ), and hence (12)

(a + c, b + d) θ (a0 + c0 , b0 + d0 ).

If (a, b) is one of the pairs {(0, 0), (1, 0), (0, 1)}, then (11) says that (a, b) = (a0 , b0 ) and consequently (12) also holds. Finally, if (a, b), (c, d) are both members of the set {(0, 0), (1, 0), (0, 1)}, we reason as before and obtain the same conclusion. All this proves our claim, and therefore we have that B ∈ F E(A∗ ) but B ∈ / EF (A∗ ), since no quotient of B can have as underlying algebra the additive group of integers. A similar counterexample can be found that proves F ∗ 6= F ∗ L. The previous remark says that the universal atomic class generated by a class K cannot be obtained by adding ER to the classical operator HSP that generates universal atomic classes in presence of the equality symbol; so far, this had constituted the only necessary modification with respect to the model theory developed by Mal’cev. Also, since F ∗ 6= F ∗ L, the class F ∗ S ∗ P ∗ (K∗ ) does not necessarily coincide with (KV )∗ . In view of this, two interesting issues arise naturally: to determine sufficient conditions for the class K to satisfy the equalities KV = ERHSP (K) and (KV )∗ = F ∗ S ∗ P ∗ (K∗ ). An answer to these problems can be found in [10]. Just notice that none of the inequalities F ∗ S ∗ P ∗ ≤ F ∗ ESP and F ∗ ESP ≤ F ∗ S ∗ P ∗ seem to hold in general. Acknowledgements. I would like to thank Professor Don Pigozzi for helping me develop my ideas through many stimulating conversations, and to Professors Janusz Czelakowski, Ramon Jansana and Josep M. Font for their comments on a draft version of this work. References [1] [2]

[3] [4] [5]

W. Blok and D. Pigozzi, Algebraizable logics, Memoirs of the AMS 396 (1989), 78pp. , Algebraic semantics for Universal Horn Logic without equality, in A. Romanowska, J.D. Smith (eds.), “Universal Algebra and quasi-groups”, Heldermann Verlag, Berlin, 1992, pp. 1–56. S. Burris and H.P. Sankappanavar, “A course in Universal Algebra”, Graduate Texts in Mathematics, Vol. 78, Springer-Verlag, New York, 1981. C.C. Chang and H.J. Keisler, “Model Theory” (3rd edition), Studies in Logic and the Foundations of Mathemantics, Vol. 73, North-Holland, Amsterdam, 1990. J. Czelakowski, Reduced products of logical matrices, Studia Logica 39 (1980), 19–43.

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, Some theorems on structural entailment relations, Studia Logica 42 (1983), 417– 430.

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, Personal communication, 1994. J. Czelakowski, W. Dziobiak, Another proof that ISPr is the least quasivariety containing K, Studia Logica 39 (1980), 343–345. R. Elgueta, “Algebraic Model Theory for Languages without Equality”, PhD Thesis, 1994. , “Algebraic Study of Structure Classes over Equality-free Languages I: Subdirect Representation Theory”, Preprint, 1995. , “Algebraic Study of Structure Classes over Equality-free Languages II: Freeness and Presentations”, Preprint, 1996. R. Elgueta, R. Jansana, “Definability of Leibniz Equality”, Preprint, 1995. G. Gr¨ atzer, “Universal Algebra” (2nd edition), Springer-Verlag, New York, 1979. G. Gr¨ atzer, H. Lasker, A note on the implicational class generated by a class of structures, Canadian Math. Bull. 16 (1973), 603–605. A.I. Mal’cev, “The metamathematics of algebraic systems”, Collected Papers: 1936-1937, Studies in Logic and the Foundations of Mathematics, Vol. 66, North-Holland, Amsterdam, 1971. , “Algebraic Systems”, Die Grundlehren der Mathematischen Wissenschaften, Band 152, Springer-Verlag, Berlin, 1973. W. Wechler, “Universal Algebra for Computer Scientists”, E.A.T.C.S. Monographs, Vol. 25, Springer-Verlag, New York, 1992. G. Zubieta, Clases aritm´ eticas definidas sin igualdad, Bol. Soc. Mat. Mexicana 2 (1957), 45–53.