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Example 2. In the variety of Boolean [resp. Heyting] algebras, which alge- braizes classical [resp. intuitionistic] propositional logic, (1) takes the form (x → y ≈ 1 & y → x ≈ 1) ⇐⇒ x ≈ y. In the variety of commutative residuated lattices [14], which algebraizes a rich fragment of linear logic, (1) is most naturally instantiated as ((x → y) ∧ 1 ≈ 1 & (y → x) ∧ 1 ≈ 1) ⇐⇒ x ≈ y. 

Given K, τ and ρ as in Definition 1, we can construct a logic ⊢K,τ for which K is the equivalent algebraic semantics, as follows. It is convenient here to fix a proper class Var of variables for the entire discussion. For each set X ⊆ Var, a term ϕ over X in the signature of K is declared X X a ⊢K,τ -consequence of a set Γ of such terms (written as Γ ⊢K,τ ϕ) provided that the following is true: for any homomorphism h from the absolutely free algebra T (X) over X to any member of K, the kernel of h contains

τ (ϕ) := {hδi(ϕ), εi(ϕ)i : i ∈ I} whenever it contains τ [Γ] := Sγ∈Γ τ (γ). (This criterion is abbreviated as τ [Γ] |=K τ (ϕ).) (2) X Thus, ⊢K,τ is a binary relation from the power set of T (X) to T (X). For any two sets X,Y ⊆ Var, with Γ ∪ {ϕ} ⊆ T (X) ∩ T (Y ), it can be X Y verified that Γ ⊢K,τ ϕ iff Γ ⊢K,τ ϕ. It therefore makes sense to write X Γ ⊢K,τ ϕ if there exists a set X ⊆ Var such that Γ ⊢K,τ ϕ. X Technically, ⊢K,τ is the family of relations ⊢K,τ indexed by the subsets X of Var. It has the following properties for any sets X,Y ⊆ Var, any Γ ∪ Ψ ∪ {ϕ}⊆ T (X) and any homomorphism h: T (X) −→ T (Y ), where we abbreviate ⊢K,τ as ⊢ : (i) if ϕ ∈ Γ, then Γ ⊢ ϕ; (ii) if Γ ⊢ ψ for all ψ ∈ Ψ, and Ψ ⊢ ϕ, then Γ ⊢ ϕ; (iii) if Γ ⊢ ϕ, then h[Γ] ⊢ h(ϕ). For present purposes, (i)–(iii) are the defining properties of logics (over Var) in general. Notice that ⊢K,τ is defined and is a logic for any class K of similar algebras and any set τ of pairs of unary terms in its signature (regardless of ρ and (1)). We say that a logic ⊢ is finitary if it has the following additional property: (iv) whenever Γ ⊢ ϕ, then Γ′ ⊢ ϕ for some finite Γ′ ⊆ Γ. In this case, for any infinite set X ⊆ Var, the logic ⊢ is determined by its restriction to terms over X, and (iii) need only be stipulated for the endomorphisms h of T (X). A finitary logic is usually (and can always be) specified by a formal system F of axioms and finite inference rules, where the natural deducibility relation of F satisfies (i)–(iv). PREVARIETIES OF LOGIC 3

Definition 3. If a logic ⊢ has the form ⊢K,τ for some prevariety K and transformers τ and ρ, where K satisfies (1), then K is said to algebraize ⊢, and ⊢ is said to be algebraizable. 1  Under these conditions, K is called the equivalent algebraic semantics of ⊢, because it is the only prevariety that algebraizes ⊢; the transformers are essentially unique as well, cf. [4, Thm. 2.15]. A quasivariety that algebraizes two different logics (via different transformers τ and a common ρ) is exhib- ited in [4, Sec. 5.2], along with several non-algebraizable finitary logics. An algebraizable logic is said to be finitely algebraizable if, in its alge- braization, the transformer ρ can be chosen finite, i.e., the index set J in (1) can be kept finite. That happens, for instance, whenever the equivalent algebraic semantics K is a quasivariety. Dually, if an algebraizable logic ⊢ is finitary, then a finite choice of τ (i.e., of I) is possible in (1). These facts and the next lemma are proved, for instance, in [13, Lem. 3.37]. Lemma 4. Let K be a quasivariety of logic, with transformers τ and ρ as in (1). Then τ can be chosen finite iff ⊢K,τ is finitary. (The finitely algebraizable finitary logics coincide with the original ‘alge- braizable logics’ of Blok and Pigozzi [4]. For cases excluded by their defini- tion, see [17, 27].) Notice that (1) is valid in a class K of similar algebras iff it is valid in the prevariety ISP(K) generated by K. Our focus on prevarieties is therefore not restrictive. In contrast with the case of quasivarieties, it is not prov- able in the class theory NBG (with choice) that every prevariety has an axiomatization involving only a set of variables [1]. Papers dealing with the algebraization of logics over proper classes of variables include [2, 11, 23]. Remark 5. A variety K that satisfies f(x,x,...,x) ≈ x for each of its basic operation symbols f is said to be idempotent. In this case, if (1) is valid in K, then K satisfies δi(x) ≈ x ≈ εi(x) for all i ∈ I, making the left hand side of (1) true on any interpretation of x,y in any member of K. In view of the right hand side of (1), this forces K to be trivial. Thus, no nontrivial idempotent variety is a variety of logic. In particular, a nontrivial variety of lattices cannot be a variety of logic, as it is idempotent. More strikingly, al- though De Morgan lattices are a modest generalization of Boolean algebras, in essentially the same signature, the variety of De Morgan lattices (which is not idempotent) also fails to be a variety of logic [12]. 

2. Category Equivalences A class K of similar algebras can be treated as a concrete category, the morphisms being the algebraic homomorphisms between members of K.

1 This implies that ⊢ and the full equational consequence relation |= K of K are essen- tially interchangeable (i.e., the interpretation given by (2) is invertible), but we shall not dwell further on the interpretations here. See [13] for additional motivation. 4 TOMMASOMORASCHINIANDJAMESG.RAFTERY

Termwise equivalent classes are then categorically equivalent, but not con- versely. When a prevariety K algebraizes a logic ⊢, we sometimes discover signif- icant features of ⊢ via ‘bridge theorems’ of the form ⊢ has metalogical property P iff K has algebraic property Q. (3) Examples include connections between metalogical interpolation properties and algebraic amalgamation properties [11], between definability theorems and the surjectivity of suitable epimorphisms [2, 23], and between deduction- like theorems and congruence extensibility properties [3, 5, 10]. As it happens, the algebraic properties Q alluded to here are categorical, i.e., they persist under category equivalences between classes K of the kind to which (3) applies. In such cases, if we wish to establish P for ⊢, we are not forced to prove Q in K directly; it suffices to prove Q in an equally suitable class M that is categorically equivalent to K. 2 That being so, and in view of Remark 5, it is natural to ask which pre- varieties are categorically equivalent to prevarieties of logic. We proceed to prove that this is true of every prevariety. Definition 6. Given an algebra A and n ∈ ω = {0, 1, 2,... }, we denote by Tn(A) the set of all n-ary terms in the signature of A. For n> 0, the n-th matrix power of A is the algebra [n] n n A := hA , {mt : t ∈ Tkn(A) for some positive k ∈ ω}, n n k n where for each t = ht1,...,tni ∈ Tkn(A) , we define mt : (A ) −→ A as n follows: if aj = haj1, . . . , ajni∈ A for j = 1,...,k, then A mt(a1, . . . , ak)= hti (a11, . . . , a1n, . . . , ak1, . . . , akn) : 1 ≤ i ≤ ni. (Roughly speaking, therefore, the basic operations of A[n] are all conceivable operations on n-tuples that can be defined using the terms of A.) For 0 < n ∈ ω, the n-th matrix power of a class K of similar algebras is the class K[n] := I{A[n] : A ∈ K}. 

Applications of the matrix power construction in universal algebra range from the algebraic description of category equivalences and adjunctions [21, 22] to the study of clones [24], Maltsev conditions [29, 15], and finite algebras [18]. Matrix powers are also the basis for ‘twist-product’ constructions and product representations; see for instance [9].

2 The value of this observation lies not only in the hope that M can be chosen simpler or ′ better-understood than K, but also in the possibility that M algebraizes a logic ⊢ , different from ⊢ (perhaps in a different signature). In that situation, a category equivalence F between M and K carries positive and negative results from one whole family of logics to another. This is because, in the case of varieties for instance, F induces an isomorphism between the respective sub(quasi)variety lattices of M and K, along which categorical properties can still be transferred. And the subquasivarieties of K [resp. M] algebraize the ′ extensions of ⊢ [resp. ⊢ ], with subvarieties corresponding to axiomatic extensions. PREVARIETIES OF LOGIC 5

Theorem 7. (cf. [21, Thm. 2.3]) Let K be a class of similar algebras and n a positive integer. Then K[n] is a class of similar algebras, which is categor- ically equivalent to K. Moreover, if K is a prevariety [resp. a quasivariety; a variety], then so is K[n].

Proof. It is not difficult to see that the functor (·)[n] : K → K[n] sending alge- bras A ∈ K to A[n] ∈ K[n] and replicating homomorphisms componentwise O S P P H is a category equivalence. And for each class operator among , , u, , it is easily verified that K is closed under O iff the same is true of K[n].  We can now prove the main result of this section. Theorem 8. Let K be any prevariety. Then K is categorically equivalent to a prevariety of logic, i.e., to the equivalent algebraic semantics M of some algebraizable logic ⊢. Moreover, we can choose M in such a way that the transformers τ and ρ in (1) are finite, and we can arrange that M is a [quasi]variety if K is. If K is a quasivariety, then ⊢ can be chosen finitary. Proof. Let M be the matrix power K[2]. By Theorem 7 and Lemma 4, we need only prove that M satisfies (1) for some finite transformers τ , ρ (in which case ⊢M,τ can serve as ⊢). Now each member of M has basic binary operations → and ←, and a basic unary operation ✷ such that, for all A ∈ K and a, b, c, d ∈ A, A[2] ha, bi→ hc, di = ha, ci = hπ1(a, b, c, d), π3(a, b, c, d)i; A[2] ha, bi← hc, di = hb, di = hπ2(a, b, c, d), π4(a, b, c, d)i; A[2] ✷ ha, bi = hb, ai = hπ2(a, b), π1(a, b)i, where πk(z1,...,zn) := zk whenever 1 ≤ k ≤ n ∈ ω. These are indeed basic operations for M, because projections are term functions of A. For every A ∈ K and a, b, c, d ∈ A, we have ha, bi = hc, di iff a = c and b = d iff ha, ci = hc, ai and hb, di = hd, bi A[2] A[2] A[2] iff ha, bi→ hc, di = ✷ (ha, bi→ hc, di) and A[2] A[2] A[2] ha, bi← hc, di = ✷ (ha, bi← hc, di)
. This implies that the following formula is valid in M: x → y ≈ ✷(x → y)& x ← y ≈ ✷(x ← y)
⇐⇒ x ≈ y. (4) In other words, (1) becomes valid in M when we set τ (x)= {hx, ✷xi} and ρ(x,y)= {x → y, x ← y}. 3 

3 Readers who are familiar with abstract algebraic logic will notice that the reduced matrix models of ⊢M,τ are, up to isomorphism, just all hA, {ha,ai : a ∈ A}i, A ∈ K. 6 TOMMASOMORASCHINIANDJAMESG.RAFTERY

Corollary 9. The property of being the equivalent algebraic semantics of an algebraizable logic is not preserved by category equivalences between pre- varieties, quasivarieties or varieties.

Proof. This follows from Theorem 8 and Remark 5. 

3. Congruence Equations We have noted that the transformer ρ in the definition of a quasivariety of logic can be chosen finite. By a finitary variety of logic, we mean a variety of logic for which the transformer τ can also be chosen finite (i.e., ⊢K,τ is finitary—see Lemma 4). Remark 10. The finitary varieties of logic constitute a Maltsev class in the sense of [28]. Indeed, suppose

τ = {hδi, εii : i = 1,...,n} and ρ = {ρj : j = 1,...,m}. Applying Maltsev’s Lemma (cf. [8, Lem. V.3.1]) to the free 2-generated alge- bra in a variety K, we see that (1) is equivalent, over K, to the conjunction of the identities δi(ρj(x,x)) ≈ εi(ρj(x,x)) and a suitable scheme of identities

x ≈ t1(x,y, δρ(x,y), ερ(x,y))

ti(x,y, ερ(x,y), δρ(x,y)) ≈ ti+1(x,y, δρ(x,y), ερ(x,y)) (1 ≤ i < k)

tk(x,y, ερ(x,y), δρ(x,y)) ≈ y involving terms t1,...,tk, where δρ(x,y) [resp. ερ(x,y)] abbreviates

δ1(ρ1(x,y)),...,δn(ρ1(x,y)),...,δ1(ρm(x,y)),...,δn(ρm(x,y))

[resp. ε1(ρ1(x,y)),...,εn(ρ1(x,y)),...,ε1(ρm(x,y)),...,εn(ρm(x,y))].

If we leave the term symbols δi, εi, ρj,tr unspecified, then the finite con- junction above defines a strong Maltsev class, which need not be idempotent (e.g., the variety M in the proof of Theorem 8 does not satisfy ✷x ≈ x when K is a variety). There are only denumerably many such formal conjunctions, and any two of them have a common weakening of the same form, got by maximizing, for each of the letters δ, ε, ρ, t, the number of subscripted oc- currences of that letter. The finitary varieties of logic are therefore directed by the interpretability relation, whence they form a Maltsev class. 4 

Despite this observation, Theorem 8 allows us to show, by an elemen- tary argument, that finitary varieties of logic are not forced to satisfy any interesting ‘congruence equation’ in the sense of the next definition. This contrasts with the fact that the most familiar varieties of logic—the ‘point- regular’ varieties—are congruence modular and congruence n-permutable for a suitable finite n [16]. (They include the varieties in Example 2.)

4 Alternatively, it can be verified that the non-indexed product of two finitary varieties of logic is a finitary variety of logic; see [19, 25, 28] for the pertinent characterizations. PREVARIETIES OF LOGIC 7

Definition 11. A congruence equation is a formal equation in the binary symbols ∧, ∨ and ◦. It is satisfied by an algebra A if it becomes true whenever we interpret the variables of the equation as congruence relations of A, and for arbitrary binary relations α and β on A, we interpret α ∧ β, α ∨ β and α ◦ β as α ∩ β,ΘA(α ∪ β) and the relational product, respectively. (Here, ΘA stands for congruence generation in A.) A congruence equation is satisfied by a class of algebras if it is satisfied by every member of the class. It is nontrivial if some algebra fails to satisfy it.  Because ◦ is not generally a binary operation on congruences, we associate with each algebra A another algebra Rel(A) = hRel(A); ∩, ∨, ◦i, where Rel(A) is the set of all binary relations on A, and A α ∨ β := Θ (α ∪ β) for all α, β ∈ Rel(A). The congruence lattice of A is therefore a subalgebra of the ∩, ∨ reduct of Rel(A). Given α, β ∈ Rel(A), we also define α ⊗ β = {hha, bi, hc, dii : ha, ci∈ α and hb, di∈ β}∈ Rel(A2). For congruences α, β of A, it is well known that α ⊗ β is a congruence of the algebra A2, but in fact it is also a congruence of A[2]. This follows straightforwardly from the definitions of A[2] and α ⊗ β. Recall that a polynomial of an algebra hA; F i is a term function of the algebra hA; F ∪ F0i, where F0 consists of the elements of A, considered as nullary basic operations. (Of course, we arrange first that A ∩ F = ∅.) Lemma 12. Let A be an algebra. Then λ: α 7→ α⊗α defines an embedding of Rel(A) into Rel(A[2]), which maps congruences to congruences. Proof. It is easily verified that, as a function from Rel(A) to Rel(A[2]), λ is injective, ∩-preserving and ◦-preserving. Let α, β ∈ Rel(A). We have already mentioned that λ preserves congruencehood, from which it follows that λ(α) ∨ λ(β) ⊆ λ(α ∨ β). It remains to prove the reverse inclusion. Accordingly, let hha, bi, hc, dii ∈ λ(α ∨ β), so ha, ci, hb, di ∈ α ∨ β. The closure operator ΘA (on the power set of A2) is algebraic, so there exist

he1, g1i,..., hem, gmi∈ α and hem+1, gm+1i,..., he2m, g2mi∈ β (5) A with ha, ci, hb, di ∈ Θ {he1, g1i,..., hem, gmi, hem+1, gm+1i,..., he2m, g2mi} (where m is finite). By Maltsev’s Lemma, therefore, there are finitely many 4m-ary polynomials p1,...,pk and q1,...,qk of A such that

a = p1(e1,...,e2m, g1,...,g2m)

pi(g1,...,g2m, e1,...,e2m)= pi+1(e1,...,e2m, g1,...,g2m)

pk(g1,...,g2m, e1,...,e2m)= c ;

b = q1(e1,...,e2m, g1,...,g2m)

qi(g1,...,g2m, e1,...,e2m)= qi+1(e1,...,e2m, g1,...,g2m)

qk(g1,...,g2m, e1,...,e2m)= d 8 TOMMASOMORASCHINIANDJAMESG.RAFTERY

for i = 1,...,k − 1. For each i ∈ {1,...,k − 1}, the rules

pi(x1,...,x4m):= pi(x1,x3,...,x2m−1,x2,x4,...,x2m, b x2m+1,x2m+3,...,x4m−1,x2m+2,x2m+4,...,x4m);

qi(x1,...,x4m):= qi(x1,x3,...,x2m−1,x2,x4,...,x2m, b x2m+1,x2m+3,...,x4m−1,x2m+2,x2m+4,...,x4m);

ti(z1,...,z2m) := hpi(π1(z1),π2(z1),...,π1(z2m),π2(z2m)), b qi(π1(z1),π2(z1),...,π1(z2m),π2(z2m))i b define two new 4m-ary polynomials of A and a 2m-ary polynomial ti of [2] 2 A , such that for any hs1, u1i,..., hs2m, u2mi∈ A , the respective first and second co-ordinates of ti(hs1, u1i,..., hs2m, u2mi) are

pi(s1,s2,...,sm, u1, . . . , um,sm+1,...,s2m, um+1, . . . , u2m)

and qi(s1,s2,...,sm, u1, . . . , um,sm+1,...,s2m, um+1, . . . , u2m). It follows that

ha, bi = t1(he1, em+1i,..., hem, e2mi, hg1, gm+1i,..., hgm, g2mi);

ti(hg1, gm+1i,..., hgm, g2mi, he1, em+1i,..., hem, e2mi)

= ti+1(he1, em+1i,..., hem, e2mi, hg1, gm+1i,..., hgm, g2mi);

tk(hg1, gm+1i,..., hgm, g2mi, he1, em+1i,..., hem, e2mi)= hc, di for i = 1,...,k − 1, whence A[2] hha, bi, hc, dii ∈ Θ {hhe1, em+1i, hg1, gm+1ii,..., hhem, e2mi, hgm, g2mii}. (6) Now let j ∈ {1,...,m}. By (5),

hhej , eji, hgj , gjii ∈ λ(α) and hhem+j, em+j i, hgm+j , gm+j ii ∈ λ(β). (7)

[2] For the basic operation →A in the proof of Theorem 8, we have A[2] hej , em+ji = hej, eji→ hem+j, em+j i A[2] hgj , gm+ji = hgj, gji→ hgm+j , gm+ji,

so hhej, em+j i, hgj , gm+j ii ∈ λ(α) ∨ λ(β), by (7). Then, since j ∈ {1,...,m} was arbitrary, (6) yields hha, bi, hc, dii ∈ λ(α) ∨ λ(β), as required.  Corollary 13. If a congruence equation fails in an algebra A, then it fails in A[2].

This allows us to prove the main result of this section: Theorem 14. Every nontrivial congruence equation fails in some finitary variety of logic, i.e., in a variety that is the equivalent algebraic semantics of some finitely algebraizable finitary logic. PREVARIETIES OF LOGIC 9

Proof. Each nontrivial congruence equation fails in some variety K, hence also in K[2] (by Corollary 13), which is itself a variety (by Theorem 7). And K[2] is a finitary variety of logic, by the proof of Theorem 8. 

A finitary variety of logic satisfying no nontrivial congruence equation in the signature ∧, ∨ (excluding ◦) was exhibited in [6]. The stronger fact that this variety satisfies no nontrivial idempotent Maltsev condition was pointed out in [7, Sec. 10.1], using [20, Thm. 4.23]. Theorem 14 does not follow from these observations and general results of universal algebra, however, because it is not evident that every nontrivial congruence equation (in the full signature ∧, ∨, ◦) entails a nontrivial idempotent Maltsev condition, as opposed to a weak Maltsev condition. (This corrects an impression left in the last lines of [7, p. 647], and in [26].) For more on the general connections between these notions, see [20].

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Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodarenskou´ veˇˇz´ı 2, 182 07 Prague 8, Czech Republic. E-mail address: [email protected]

Department of Mathematics and Applied Mathematics, University of Pre- toria, Private Bag X20, Hatfield, Pretoria 0028, South Africa E-mail address: [email protected]