Almost structural completeness; an algebraic approach

Annals of Pure and Applied Logic 167 (2016) 525–556

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Annals of Pure and Applied Logic www.elsevier.com/locate/apal

Almost structural completeness; an algebraic approach Wojciech Dzik a , Michał M. Stronkowski b,∗,1 a

Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland Faculty of Mathematics and Information Sciences, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland b

a r t i c l e

i n f o

Article history: Received 20 August 2014 Received in revised form 5 September 2015 Accepted 23 December 2015 Available online 24 March 2016 MSC: 08C15 03G27 03B45 06E25 Keywords: Almost structural completeness Structural completeness Quasivarieties Axiomatization Modal normal logics Varieties of closure algebras

a b s t r a c t A deductive system is structurally complete if all of its admissible inference rules are derivable. For several important systems, like the modal logic S5, failure of structural completeness is caused only by the underivability of a passive rule, i.e., a rule whose premise is not unifiable by any substitution. Neglecting passive rules leads to the notion of almost structural completeness, that means, to the derivability of admissible non-passive rules. We investigate almost structural completeness for quasivarieties and varieties of general algebras. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: the one expressed with finitely presented algebras, and the one expressed with subdirectly irreducible algebras. The next one is restricted to quasivarieties with the finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Some connections with exact and projective unification are included. Secondly, examples of almost structurally complete varieties are provided. Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of the modal logic S4. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification is not unitary there. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The paper is a contribution to the study of admissible inference rules from an algebraic perspective. Roughly speaking, an inference rule is admissible for a deductive system (or a logic) S if it does not produce * Corresponding author. 1

E-mail addresses: [email protected] (W. Dzik), [email protected] (M.M. Stronkowski). The work was supported by the Polish National Science Centre grant no. DEC-2011/01/D/ST1/06136.

http://dx.doi.org/10.1016/j.apal.2016.03.002 0168-0072/© 2016 Elsevier B.V. All rights reserved.

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new theorems when added to S [43,45,46,48,49,66,78]. Clearly, every derivable rule is admissible but the converse does not need to hold. The concept of admissibility was formally introduced by Lorenzen in [55] but it appeared much earlier. A well known admissible rule is the cut rule of Gentzen’s sequent systems for classical and intuitionistic logics. Note that for a given deductive system admissibility depends only on its theorems. Derivability does not share this property. In [41] a deductive system is constructed with the set of classical tautologies as the set of theorems but in which even the Modus Ponens rule is not derivable. The study of admissibility for non-classical logics was stimulated by Friedman’s problem. It asks whether the admissibility of rules is decidable for the intuitionistic logic [33]. This problem was solved by Rybakov in [78]. But admissibility was further investigated in intuitionistic and modal logics. New horizons were opened when Ghilardi successfully applied unification to admissibility problems. Ghilardi not only provided a new and elegant solution to Friedman’s problem [34,35] but also gave new effective tools for studying admissibility of rules and linked logic and computer science. In order to present the motivation for introducing almost structural completeness and for studying admissibility, we recall some notions from propositional logic. Let L be a propositional language, i.e., a set of logical connectives with ascribed arities, and let Form be the algebra of formulas in L over a denumerable set of variables. An (inference) rule is a pair from Pfin (Form) × Form, written as Φ/ϕ, where Pfin (Form) is the set of all finite subsets of Form. By a deductive system we mean a pair S = (Form, ), where  is a (finitary structural) consequence relation, this is, a set of rules satisfying the following postulates2 [19,30, 31,70,72,75,83,84]: for all Φ, Ψ ∈ Pfin (Form) and ϕ ∈ Form we have • if ϕ ∈ Φ, then Φ  ϕ, • if for all ψ ∈ Ψ, Φ  ψ, and Ψ  ϕ, then Φ  ϕ, • for every substitution σ (i.e., an endomorphism of Form), if Φ  ϕ, then σ(Φ)  σ(ϕ). The set Th(S) = {ϕ ∈ Form | ∅  ϕ} is the set of theorems of S. A logic is a subset of Form which is closed under substitution and some default rules. For instance, for intermediate logics this is the Modus Ponens rule and for normal modal logics these are the Modus Ponens and the Necessitation rules. With a logic L (with a set R of default rules) one can associate the deductive system S = (Form, ) such that L = Th(S) and  is minimal with respect to R ⊆ . A basis or an axiomatization of a deductive system S (or of a logic with a default set R of rules) is a pair (A, R) (or just a set A), where A ⊆ Th(S) and R ⊆  are such that  is the smallest consequence relation containing R ∪ {∅/α | α ∈ A}. It means that Φ  ϕ iff there is a proof or derivation (in some strict sense) from A ∪ Φ for ϕ by means of the rules from R. Given a basis (A, R) of a deductive system S one is interested whether a formula ϕ is in Th(S), i.e., whether there is a proof of ϕ from A with the application of the rules from R. Thus the issue of the size of such proofs arises. Proofs of theorems may be shortened by allowing new rules. Such extension of R may be done in two ways: 1. by adding derivable rules; 2. by adding admissible but non-derivable rules. A rule is derivable if it is in . And a rule Φ/ϕ is admissible if for every substitution σ we have σ(ϕ) ∈ Th(S) whenever σ(Φ) ⊆ Th(S).

2 We adopt the slightly modified definition from [31]. However it is also a common practice to use the term “deductive system” for the basis of deductive system in our sense.

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Admissibility is more elusive than derivability and its verification may be a challenging task [78]. Deductive systems (and logics) for which all admissible rules are derivable are called structurally complete (SC for short). A rule Φ/ϕ is passive if for every substitution σ the set σ(Φ) is not contained in Th(S); this means that Φ is not unifiable. Such rule is admissible but cannot be used in any proof of a theorem of S. (Note, however that there are deductive systems like Belnap’s four-valued logic FDE which has no theorems and hence in which every passive rule is admissible.) There are many deductive systems that are not SC but in which the only admissible non-derivable rules are passive. So in these deductive systems proofs of theorems cannot be shortened by means of the method (2). Such systems are called almost structurally complete (ASC for short). Researchers studying admissible rules and unification focus usually on determining the decidability, axiomatization and complexity of admissible rules for a given variety or logic. Almost structural completeness reduces these questions significantly to questions of derivability and unifiability. The following example of modal logics shows an advantage of ASC over SC. Let L be a modal normal logic with a basis A (here R consists of the Modus Ponens and the Necessitation rules). Recall that L has an adequate algebraic semantics given by a variety V of modal algebras (see Section 7 for definitions of modal and closure algebras). A formula ϕ(¯ x) holds in a modal algebra M provided M |= (∀¯ x) ϕ(¯ x) ≈ 1, and, a rule ϕ1 (¯ x), . . . , ϕn (¯ x)/ϕ(¯ x) (we adopt a common convention and drop the curly brackets) holds in M if the quasi-identity (∀¯ s)[ϕ1 (¯ x) ≈ 1 ∧ · · · ∧ ϕn (¯ x) ≈ 1 → ϕ(¯ x) ≈ 1] holds in M. Then a formula belongs to L iff it holds in all algebras from V, and similarly a rule is derivable iff it holds in all algebras from V.

Fig. 1. The modal algebras 2 and S2 .

Assume that the algebras 2 and S2 , depicted in Fig. 1, are in V. The algebras 2 and S2 are closure algebras with only the top and the bottom elements closed (and open). Then L, which corresponds to V, cannot be SC. Indeed, the rule P2 3p, 3¬p / ⊥ is not valid in S2 . Therefore it is not derivable. However, P2 is admissible. Indeed, 2 satisfies the sentence (∀y)[3y ≈ 1 ∨ 3¬y ≈ 1]. Hence 2 |= (∀¯ x)[3ϕ(¯ x) ≈ 1 ∧ 3¬ϕ(¯ x) ≈ 1] and, consequently, we have {3ϕ(¯ x), 3¬ϕ(¯ x)}  L, for every modal formula ϕ(¯ x). Thus P2 is passive and, therefore, admissible. There are many normal modal logics of this kind which are ASC. For example, the modal logic S5 (the logic of equivalence relations) is ASC. Its algebraic semantics is given by the variety of monadic algebras introduced by Halmos [38,39]. This variety will be considered in Example 7.8. Also the modal logic S4.3 is ASC [27]. It is even hereditary ASC in the sense that all of its normal extensions are ASC [27] (see also Example 7.10). It appears that from the single rule P2 all passive rules may be derived in every normal modal logics extending S4 [79]. Still, one can ask whether shifting from SC to ASC brings a significant number of new deductive systems. We call distinguishing non-SC deductive systems among ASC ones the SC versus ASC problem. A general solution was given in [15]. In the same paper it was shown that if L is a logic with a ground (constant) formula which is a theorem and with a ground formula which is not a theorem, and which has an adequate semantics given by a discriminator variety, then L is SC iff it is maximal. All such logics are ASC. It follows

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that all (continuum many) varieties (and logics corresponding to them) of relation algebras are ASC, but only four of them are SC (three minimal and one trivial). Similarly, all (continuum many) varieties of two-dimensional cylindric algebras are ASC, but only three of them are SC (two minimal and one trivial). The SC versus ASC problem was also solved for 3-element groupoids [66]. It appears that 2676 of all 3330 (up to isomorphism) generate SC quasivarieties and additional 254 of them generate ASC quasivarieties. In this paper we present the solution of this problem for varieties of closure algebras (Proposition 8.7). The notion of ASC is studied here by means of algebraic semantics. Recall that every algebraizable deductive system S has an adequate semantics which is a quasivariety Q of algebras [7]. In particular: logical connectives become basic operations, formulas become terms, theorems of S correspond to identities true in Q, derivable rules of S correspond to quasi-identities true in Q, and admissible rules of S correspond to quasi-identities valid in free algebras for Q. Therefore the property of being ASC may be formulated for quasivarieties. In the first part of the paper (Sections 3–6) we develop a general theory of the ASC property for quasivarieties. We present various characterizations of ASC quasivarieties. Two of them are general: the first expressed with finitely presented algebras (Theorem 3.1 Condition (5)), and the second expressed with subdirectly irreducible algebras (Theorem 3.1 Condition (3)). The third characterization is restricted to quasivarieties with the finite model property and equationally definable principal relative congruences. There the condition is verifiable on finite subdirectly irreducible algebras (Theorem 6.1). We would like to note that the condition for being ASC expressed with finitely presented algebras has connection to unification theory (see Section 5). But the condition expressed with subdirectly irreducible algebras is purely algebraic and probably could not be discovered without algebraic tools. In the second part of the paper (Sections 7 and 8) we illustrate theoretical considerations by showing how our results may be used to establish ASC for particular varieties. Until now the most common method for proving ASC was by an application of projective unification, see e.g. [25–28], (however, see [66,14] for exceptions). In contrast, we use algebraic tools like subdirectly irreducible algebras and free algebras. We put an emphasis on varieties of closure algebras. They constitute an adequate semantics for transitive reflexive normal modal logics, i.e., for normal extensions of the modal logic S4. The main result here is the description of an infinite family of ASC, but not SC, varieties. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence they do not have unitary (unitary exact) unification (Theorem 8.12). In particular, a verification of ASC for them could not be obtained by means of projective unification. The idea behind the construction is the following one. We consider a variety U of SC closure algebras without projective unification. We “spoil it a bit” by taking the varietal join U ∨ W with a non-minimal nontrivial variety W of monadic algebras. Non-minimal nontrivial varieties of monadic algebras are known to be ASC but not SC. In order to prove that such join is still ASC we have to develop the theory of closure algebras. In particular, on the way, we show that an ASC variety of closure algebras is SC iff it satisfies the McKinsey identity (Proposition 8.7). This solves SC versus ASC problem for varieties of closure algebras. Moreover, we describe free algebras for U ∨W by means of free algebras for U and W, where U is a variety of McKinsey algebras and W is a variety of monadic algebras (Proposition 8.2). As a by-product we obtain that every finitely generated algebra from U ∨ W is isomorphic to a direct product of algebras from U and W (Corollary 8.3). Finally, for U we can take the smallest modal companions of Levin or Medvedev varieties of Heyting algebras described in Example 7.11. These varieties characterizes Levin or Medvedev intermediate logics which are known to be SC [71] and do not have projective unification [23]. Our results may be applied to the axiomatization problem for quasivarieties and to establishing a basis of admissible rules of a deductive system. Indeed, if a quasivariety Q (a deductive system S) is ASC, then an axiomatization of each of its subquasivarieties containing all free algebras for Q (the extension of S obtained by adding some admissible rules) may be obtained by adding passive quasi-identities (rules). In particular, an axiomatization of the quasivariety generated by all free algebras for Q (the extension of S obtained by adding

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all admissible rules) may be obtained by adding all passive quasi-identities (all passive rules), see Section 4. This fact was used in [28], where the analysis of passive rules lead to description of the lattice of all deductive systems extending the modal logic S4.3. Also bases of admissible rules consisting of passive rules were given for some ASC logics, see e.g. [49] for finite valued Łukasiewicz logics and [65] for De Morgan lattices. Historical notes The notion of SC was introduced by Pogorzelski in [69] and then investigated by many authors. The reader may consult the monograph [70] and references therein for older results concerning SC property. Let us here recall recent works on SC: [67] on varieties of positive Sugihara monoids, [68] on substructural logics, [63] on fragments of R-mingle, [18] on fuzzy logics, [36] on quasivarieties of MV-algebras, [81] on some fragment of the intuitionistic logic, [52] on BCK logic, [15] on semisimple quasivarieties and discriminator varieties, and [74] which contains general considerations from abstract algebraic logic perspective and results for some non-algebraizable deductive systems. Although the SC property was often investigated algebraically, there are few papers on it for algebras not connected to logic. Among them the paper of Bergman [3] deserves special attention. In particular, he formulated the condition for a quasivariety Q to be SC. It says that Q must be generated by its free algebras. For specific algebras, SC was investigated for lattices in [47] and for modules in [42]. ASC appeared for the first time, though under different name in [24] as an application of projective unification. Projective unification and thus ASC was established for some varieties and logics. Probably the most prominent examples are discriminator varieties. This includes varieties of e.g. Boolean algebras, monadic algebras, rings satisfying xm ≈ x for a finite m > 1, MVn -algebras, n-valued Post algebras, cylindric algebras of dimension n, all for finite n [12, Theorem 3.1]. For intuitionistic logic it was shown that every extension of Gödel–Dummett logic LC has projective unification [86]. For a normal extension L of the modal logic S4 it was proved that L has projective unification iff L is an extension of S4.3 [27, Corollary 3.19], see also [44, Theorem 5]. Projective unification was also established for some others modal logics not extending S4 [24], for k-potent extensions of basic fuzzy logic and hoops [25], and for some Fregean varieties [82]. It is worth noting that, under some mild conditions, projective unification yields ASC not only for a deductive system (quasivariety) under consideration, but also for all its extensions (subquasivarieties), see Corollary 5.5. This property is called hereditary ASC. A general investigation of the ASC property was, independently from us, undertaken in [66]. In particular, Corollary 3.2 was published there for the first time. Note however that, contrary to our paper, [66] is focused on the finitely generated case. One of the main results there is the algorithm (with an applicable software) for deciding whether a given finite family of finite algebras (of a small size) in a finite language generates an ASC quasivariety. Several variants of “completeness” for deductive systems other than SC and ASC were proposed, like maximality or Post completeness. Passive (or overflow) structural completeness [87] may be considered as complementary to ASC. A deductive system is passively structurally complete (PSC) if all of its admissible and passive rules are derivable. Clearly, a deductive system is SC iff it is ASC and PSC. The PSC property has also been investigated in the context of fuzzy logics [18]. 2. Concepts from quasivariety theory Though most deductive systems we are interested in have algebraic semantics given by varieties (they are strongly algebraizable), the proper language to deal with the SC and ASC properties and with admissibility is quasivariety theory. Indeed, we have to work with quasi-identities anyway. Therefore we will formulate our main results for quasivarieties. Let us here recall necessary notions and facts from this theory. Following [13,37,57] we call a first order sentence a quasi-identity if it is of the form (∀¯ x) [s1 (¯ x) ≈ t1 (¯ x) ∧ · · · ∧ sn (¯ x) ≈ tn (¯ x) → s(¯ x) ≈ t(¯ x)],

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where n ∈ N. We allow n to be zero, and then call the sentence an identity. It will be convenient to have a more compact notation for quasi-identities, and we will often write them in the form (∀¯ x) [ϕ(¯ x) → ψ(¯ x)], where ϕ is a conjunction of equations (i.e., atomic formulas) and ψ is an equation. We call ϕ the premise and ψ the conclusion of a quasi-identity. By a (quasi-)equational theory of a class K of algebras in the same language we mean the set of (quasi-)identities true in K. A (quasi)variety is a class defined by (quasi-)identities. Equivalently, a class of algebras in the same language is a quasivariety if it is closed under taking substructures, direct products and ultraproducts. If it is additionally closed under taking homomorphic images, it is a variety. (We tacitly assume that all considered classes contain algebras in the same language and are closed under taking isomorphic images. Also all considered class operators are assumed to be composed with the isomorphic image class operator.) A (quasi)variety is trivial if it consists of one-element algebras, and is minimal if it properly contains only a trivial (quasi)variety. We say that a class is a (quasi)variety generated by a class K if it is the smallest (quasi)variety containing K, i.e., the class defined by the (quasi-)equational theory of K. We denote this class by V(K) (Q(K) respectively). In the case when K = {A} we simplify the notation by writing V(A) (Q(A) respectively). Note that V(K) = HSP(K) and Q(K) = SPPU (K), where H, S, P, PU are the homomorphic image, subalgebra, direct product and ultraproduct class operators [13, Theorems II.9.5 and V.2.25]. Let Q be a quasivariety and A ∈ Q. A congruence α on an algebra A is called a Q-congruence provided A/α ∈ Q. The set ConQ (A) of all Q-congruences of A forms an algebraic lattice with the same meets, but possibly different joins, as in the lattice Con(A) of all congruences of A [37, Corollary 1.4.11]. We say that a (quasi)variety Q is (relatively) congruence-distributive if for every A ∈ Q the lattice ConQ (A) is distributive. A nontrivial algebra S is Q-simple if ConQ (S) has exactly two elements: the equality relation idS on S and the total relation S 2 on S. A nontrivial algebra S ∈ Q is Q-subdirectly irreducible if the equality relation on A is completely meet irreducible in ConQ (A). (In case when Q is a variety we drop the prefix “Q-”.) Let us denote the class of all Q-subdirectly irreducible algebras by QSI . The importance of QSI follows from the fact that this class determines Q. Indeed, in an algebraic lattice each element is a meet of completely meet-irreducible elements. Moreover, for A ∈ Q the lattice ConQ (A) is algebraic. Thus we have the following fact. Proposition 2.1. (See [37, Theorem 3.1.1].) Every algebra in a quasivariety Q is isomorphic to a subdirect product of Q-subdirectly irreducible algebras. In particular, Q is generated by QSI . Let G ∈ Q and X ⊆ G. We say that G is free for Q over X, and is of rank |X|, if G ∈ Q and it satisfies the following universal mapping property: every mapping f : X → A, where A is a carrier of an algebra A in Q, is uniquely extendable to a homomorphism f¯: G → A. It will be relevant in Section 8 that the above condition is equivalent to the conjunction of the following two conditions: (1) every mapping f : X → A, where A in Q, is extendable to a homomorphism f¯: G → A (a weakening of the universal mapping property) and (2) that X generates G. Elements of X are called free generators of G. If Q contains a nontrivial algebra, then it has free algebras over arbitrary non-empty sets and, in fact, they coincide with free algebras for the variety V(Q). (Note here that V(Q) is the class of all homomorphic images of algebras from Q.) Let us fix a countably infinite set of variables V = {v0 , v1 , v2 , . . .}. We denote a free algebra for Q over V by F and a free algebra for Q over Vk = {v0 , v1 , . . . , vk−1 } by F(k). In order to construct F or F(k) one may take the algebra of terms over V , or Vk respectively, and divide it by an appropriate congruence. This congruence identifies terms s(¯ v ) and t(¯ v ) iff for every A ∈ Q they determine the same term operation

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on A (in other words, when Q |= (∀¯ x)[t(¯ x) ≈ s(¯ x)]). The algebra F is an union of a chain of free algebras of finite ranks. It follows that the family of all free algebras for Q of finite rank generates the quasivariety Q(F). We notationally identify terms with elements of F that they represent. For an algebra A and a set H ⊆ A2 there exists a least Q-congruence θQ (H) on A containing H. When H = {(a, b)} we just write θQ (a, b). (When Q is a variety we also simplify the notation by dropping the subscript Q.) We say that an algebra is Q-finitely presented if it is isomorphic to F(k)/θQ (H) for some natural number k and some finite set H [37, Chapter 2]. The class of all Q-finitely presented algebras will be denoted by QFP . For a tuple x ¯ = (x0 , . . . , xk−1 ) of mutually distinct variables and a conjunction of equations ϕ(¯ x) = s1 (¯ x) ≈ t1 (¯ x) ∧ · · · ∧ sn (¯ x) ≈ tn (¯ x) let Pϕ(¯x) = F(k)/θQ ({(s1 (¯ v ), t1 (¯ v )), . . . , (sn (¯ v ), tn (¯ v ))}), where v¯ = (v0 , . . . , vk−1 ). For the definiteness of Pϕ(¯x) , let us add a formal requirement that all variables from x ¯ appear in ϕ (if necessary, we may conjunct ϕ with equalities of the form xi ≈ xi ). Note that every finitely presented algebra is isomorphic to some Pϕ(¯x) . Observe that Q satisfies a quasiidentity (∀¯ x) [ϕ(¯ x) → ψ(¯ x)] iff Pϕ(¯x) |= ψ(¯ v ) (we notationally identify variables from v¯ with their congruence classes in Pϕ(¯x) ). In particular, Q satisfies an identity (∀¯ x) [s(¯ x) ≈ t(¯ x)] iff s(¯ v ) = t(¯ v ) in F(k). We say that a Q-finitely presented algebra P is unifiable (in Q) provided that there exists a homomorphism from P into F. Every such homomorphism is called a unifier for P.3 The importance of Q-finitely presented algebras results from the following fact. Proposition 2.2. (See [37, Theorem 1.2.9, Proposition 2.1.18].) Every algebra in a quasivariety Q is isomorphic to a direct limit of Q-finitely presented algebras. Moreover, every quasivariety is closed under taking direct limits. Consequently, QFP generates Q. The following observations will be needed. Fact 2.3. Let Pϕ(¯x) be a Q-finitely presented algebra and let q = (∀¯ x) [ϕ(¯ x) → ψ(¯ x)] be a quasi-identity. Then an algebra A ∈ Q does not satisfy q if and only if there is a homomorphism h : Pϕ(¯x) → A such that A |= ψ(h(¯ x)). Fact 2.4. Let Q be a quasivariety and R be its subquasivariety. Let P be a Q-finitely presented algebra and R be an R-finitely presented algebra, both defined by the same formula. Then there is a surjective homomorphism p : P → R such that for every algebra A ∈ R every homomorphism h : P → A factors as ¯ ◦ p for some h ¯ : R → A. h We will use the following folklore fact. Fact 2.5. Let Q be a quasivariety in a finite language. Then every finite algebra in Q is Q-finitely presented. Proof. Let P be a finite algebra in Q. Take a tuple a ¯ = (a0 , . . . , ak−1 ) such that P = {a0 , . . . , ak−1 }. Let x ¯ = (x0 , . . . , xk−1 ) be a tuple of mutually distinct variables of length k. Define the set Φ(¯ x) consisting of all equations of the form ω(xi0 , . . . , xin−1 ) ≈ xin , where ω is an n-ary basic operation from the language of Q, variables are from x ¯, and ω(ai0 , . . . , ain−1 ) = ain in P. Since P is finite and the number of basic  operations is finite, Φ(¯ x) is finite. Let ϕ(¯ x) = Φ(¯ x). Then P ∼ = Pϕ(¯x) . In order to see this let us consider a homomorphism h : F(k) → P satisfying h(¯ x) = a ¯. Its existence follows from the universal mapping property. 3 Our definition of unifiers is slightly different than Ghilardi’s algebraic definition [34] in which codomains of unifiers are finitely presented projective algebras. The difference is harmless in this paper.

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Now, since the kernel of h is a Q-congruence and P |= ϕ(¯ a), the definition of Pϕ(¯x) yields that h factors through g : Pϕ(¯x) → P, where g(¯ v) = a ¯. Clearly, g is surjective. Moreover, the definition of ϕ(¯ x) implies that Pϕ(¯x) has at most k elements. Thus g must be also injective. The proof may be slightly modified in order to obtain a stronger fact: If l is the cardinality of a smallest generating set for P, then in a defining formula ϕ(¯ x) we need only l variables. 2 3. General characterizations and first observations For a quasi-identity q = (∀¯ x) [ϕ(¯ x) → ψ(¯ x)] let q ∗ = (∀¯ x) [¬ϕ(¯ x)]. We partition the set of quasi-identities true in F into two sets: the set of Q-active quasi-identities q for which q ∗ does not hold in F, and the set of Q-passive quasi-identities q for which q ∗ holds in F. Equivalently, a quasi-identity q true in F is Q-active if Pϕ(¯x) is unifiable and it is Q-passive if Pϕ(¯x) is not unifiable, where ϕ(¯ x) is the premise of q. A quasivariety Q is structurally complete (SC for short) provided that every quasi-identity which is true in F is also true in Q, in other words if Q = Q(F). A quasivariety Q is almost structurally complete (ASC for short4 ) provided that every Q-active quasi-identity holds in Q. We will also use the abbreviation ASC\SC to indicate that a considered quasivariety is ASC but is not SC. Let us start considerations by providing various conditions for quasivarieties equivalent to being ASC. We write A → B to code the supposition that there is a homomorphism from A into B. In particular, for a Q-finitely presented algebra P, P → F means that P is unifiable. Theorem 3.1. The following conditions are equivalent: 1. 2. 3. 4. 5.

Q is ASC; For every A ∈ Q, A × F ∈ Q(F); For every S ∈ QSI , S × F ∈ Q(F); For every A ∈ Q, A → F yields A ∈ Q(F); For every P ∈ QFP , P → F yields P ∈ Q(F).

Proof. The implications (2)⇒(3) and (4)⇒(5) are obvious. (1)⇒(2) Let A ∈ Q and consider a quasi-identity q true in F. We show that A × F |= q. If Q |= q, then it clearly holds since A × F ∈ Q. So suppose that Q |= q. Then, by the definition of ASC, F |= q ∗ . Thus A × F |= q ∗ , and therefore A × F |= q. x) [ϕ(¯ x) → ψ(¯ x)] and assume that F |= q and Q |= q. Then q is not valid in some (2)⇒(1) Let q = (∀¯ A ∈ Q, i.e., there is a tuple a ¯ of elements in A such that A |= ϕ(¯ a) ∧ ¬ψ(¯ a). We would like to show that q is Q-passive. Suppose that, on the contrary, F |= q ∗ . This means that there is a tuple t¯ of elements in ¯ where d¯ is the tuple of pairs of elements from a F such that F |= ϕ(t¯). Then A × F |= ϕ(d), ¯ and t¯ in the ¯ respective order. By (2), A × F |= q, and hence A × F |= ψ(d). This yields that A |= ψ(¯ a), and we obtained a contradiction. (3)⇒(2) Let A ∈ Q. By Proposition 2.1, A is isomorphic to a subdirect product of Si ∈ QSI , i ∈ I. If I = ∅, then A is trivial and A × F ∼ = F ∈ Q(F). So let us assume that I = ∅. Then F is isomorphic with the diagonal of FI , and hence A × F is isomorphic with a subalgebra of A × FI . Further, the latter is    I ∼ isomorphic to a subalgebra of i∈I Si × F = i∈I (Si × F). Thus (3) yields that A × F ∈ Q(F). 4 An alternative, and maybe better, full form of ASC would be active structural completeness as Alexander Citkin privately suggested to us.

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(2)⇒(4) Assume that there is a homomorphism h from A ∈ Q into F. Let h be the subalgebra of A × F with the carrier h. By (2) the algebra h belongs to Q(F). Since A ∼ = h, the algebra A also belongs to Q(F). (4)⇒(2) It holds since there is a homomorphism from A × F into F, namely, the second projection. (5)⇒(4) Let A ∈ Q. By Proposition 2.2, we may assume that A is a direct limit limPi of Q-finitely −→ presented algebras Pi . Let ki : Pi → A be the associated canonical homomorphisms. Assume that f : A → F. Then f ◦ ki : Pi → F, and (5) gives Pi ∈ Q(F). Since every quasivariety is closed under taking direct limits (Proposition 2.2), A belongs to Q(F). 2 The list of conditions from Theorem 3.1 is not full but we consider them as the most fundamental. In this section we formulate additional conditions equivalent to ASC which will be used in our considerations. Corollary 3.2. Let C be a subalgebra of F, e.g. F(1) or F(0) (if it exists). Then Q is ASC if and only if one of the following conditions holds. (2 ) For every A ∈ Q, A × C ∈ Q(F); (3 ) For every S ∈ QSI , S × C ∈ Q(F). Proof. For every algebra A we have A × C  A × F, and hence conditions (2) and (3) from Theorem 3.1 yield (2 ) and (3 ), respectively. For proving the converse, let us consider a homomorphism h : F → C  F. Its existence is guaranteed by the universal mapping property. Then A × F embeds into A × C × F via the mapping (a, t) → (a, h(t), t). This shows that (2 ) and (3 ) yield (2) and (3) from Theorem 3.1, respectively. 2 Remark 3.3. The equivalence of (2 ) in Corollary 3.2 with ASC was independently proved in [66, Theorem 18]. Corollary 3.4. A quasivariety Q is ASC if and only if the following condition holds. (5 ) For every P ∈ QFP , P → F yields P ∈ SP(F). Proof. Clearly (5 ) yields condition (5) from Theorem 3.1, and hence it implies ASC. For the converse consider a Q-finitely presented algebra Pϕ(¯x) and assume that it belongs to Q(F) = SPPU (F). We show that P ∈ SP(F). To this end we have to show that for each atomic formula ψ(¯ x) such that Pϕ(¯x) |= ψ(¯ v) there is a homomorphism f : Pϕ(¯x) → F such that F |= ψ(f (¯ v )). By assumption, there is a homomorphism h : Pϕ(¯x) → G, where G is some ultrapower of F, such that G |= ψ(h(¯ v )). Since G and F are elementarily equivalent, they satisfy the same quasi-identities. Thus the assertion follows from Fact 2.3. 2 From Condition (4) in Theorem 3.1 we obtain a sufficient condition under which ASC is equivalent to SC. Corollary 3.5. Suppose that every nontrivial algebra from Q admits a homomorphism into F. Then Q is ASC if and only if it is SC. Note that the assumption of Corollary 3.5 holds when F has an idempotent element, i.e., one element subalgebra. This includes cases of groups or lattices. But in quasivarieties which provide algebraic semantics for particular deductive systems we rarely have an idempotent element. This is due to the fact that for most encountered cases we have formulas for verum and falsum which correspond to two distinct constants in free algebras. However, even then Corollary 3.5 is sometimes applicable. It holds e.g. for quasivarieties of Heyting algebras (Fact 7.12) and McKinsey algebras, (Lemma 8.4). Note that the latter includes quasivarieties of

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Grzegorczyk algebras. We will return to the problem when ASC is equivalent to SC in Proposition 8.7 in the case of varieties of closure algebras. Remark 3.6. Condition (4) in Theorem 3.1 may be formulated with the use of relative antivarieties [37]. Recall that a subclass of a quasivariety Q is a Q-antivariety if it is closed under taking subalgebras, ultraproducts and homomorphic preimages from Q. Note that every Q-antivariety may be defined relative to Q by a set of anti-identities, i.e., by negative universal Horn sentences. Then (4) says that the Q-antivariety generated by F is contained in Q(F). It may be also shown that Q is ASC iff there is a nonempty Q-antivariety contained in Q(F). 4. ASC core Let Q be a quasivariety and F be its free algebra of denumerable rank. Let us consider the interval [Q(F), Q] in the lattice of subquasivarieties of Q. Notice that all quasivarieties from this interval have the same free algebras. We define the ASC core of Q to be the quasivariety defined relative to Q by all Q-active quasi-identities and denote it by ASCC(Q). It follows from the definition of ASC that ASCC(Q) is a largest ASC quasivariety in [Q(F), Q]. Note that there does not need to exist a largest ASC subquasivariety of Q. Indeed, there may exist SC subquasivariety of Q outside the interval [Q(F), Q]. A simple example is given in Example 7.1. Since ASC(Q) is defined relative to Q by Q-active quasi-identities, Q(F) is defined relative to ASC(Q) by Q-passive quasi-identities. This fact has a logical interpretation. Namely if a deductive system S is ASC, then as a basis of its admissible rules relative to S we may take the set of S-passive rules. Let us note that ASC(Q) may be defined also semantically. Proposition 4.1. For every subalgebra C of F we have ASCC(Q) = {A ∈ Q | A × C ∈ Q(F)}. Moreover, a quasivariety R from the interval [Q(F), Q] is ASC if and only if R ⊆ ASCC(Q). Proof. For convenience in this proof let us put K = {A ∈ Q | A × C ∈ Q(F)}. By Corollary 3.2, in order to prove that K = ASCC(Q) it is enough to show that K ∈ [Q(F), Q]. This means that K is a quasivariety with F as a free algebra of denumerable rank. To this end, we check its closure under S, P and PU class operators. Assume first that B  A ∈ K. Then B ×C  A ×C ∈ Q(F), and hence B ∈ K. Now assume that Ai ∈ K,    I ∼ for i ∈ I. Then, since Q(F) is closed under taking direct product, i∈I Ai × C = i∈I (Ai × C) ∈ Q(F).       I Since A × C embeds into A × C , the algebra A × C also belongs to Q(F). This i i i∈I  i i∈I i∈I proves that i∈I Ai ∈ K. For ultraproducts we argue similarly. Consider an ultrafilter U on a set I. Then     I ∼  (Ai × C)/U ∈ Q(F), and  Ai /U ∈ K. i∈I Ai U ) × C embeds into i∈I Ai /U × C /U = i∈I i∈I Hence K is a quasivariety. Moreover, F × F ∈ Q(F) shows that K has F as a free algebra of denumerable rank. Now the second statement of the proposition follows from the definition of ASC core or from Corollary 3.2. 2 5. Projective and exact unifications ASC for varieties (or logics) which are not SC in many cases was established by means of projective unification, see the historical notes in the introduction. Here we look at this and other properties from

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unification theory a bit closer (for more details see [34,64]). For an equational theory E, an E-unifier for a finite set S(¯ x) of equations is a substitution u such that (∀¯ x) u(s(¯ x)) ≈ u(t(¯ x)) belongs to E for every equation s(¯ x) ≈ t(¯ x) form S(¯ x). However, for our purposes it will be more convenient to employ a simplified version of Ghilardi’s algebraic approach [34]. Let V be the variety defined by E. Instead of working with a finite set of equations S(¯ x), we deal with the V-finitely presented algebra P S(¯x) . Then a unifier of S(¯ x)   may be identified with a unifier of P S(¯x) defined as in Section 2, i.e., with a homomorphism from P S(¯x) into F. Actually, we extend the setting to quasivarieties. There is no cost of this move, and the gain is presented in Fact 5.1. Recall that an algebra P ∈ Q is projective in Q if for every surjective homomorphism g : A → B between algebras in Q and every homomorphism f : P → B there exists a homomorphism f¯: P → A such that f = g ◦ f¯. It appears that projective algebras are retracts of free algebras. Hence the classes of projective algebras in Q and in V(Q) coincide. A quasivariety Q has projective unification if every Q-finitely presented unifiable algebra P is projective in Q. And Q has unitary unification if for every Q-finitely presented unifiable algebra P there exists a most general unifier. Recall that u : P → F is a most general unifier for P if every other unifier for P is of the form σ ◦ u for some endomorphism σ of F. Let us also recall (a fragment of) a new hierarchy of so called exact unification types. For studding admissibility it is more suitable than the classical hierarchy of unification types [64]. And indeed, it was already, although implicitly, used in [14]. For a Q-finitely presented algebra let ConF (P) be the ordered by inclusion set of congruences α of P for which P/α embeds into F. Then Q has unitary exact unification if for every Q-finitely presented unifiable algebra P the ordered set ConF (P) has exactly one minimal element (see [64, Theorem 3.7]), and Q has exact unification if for every Q-finitely presented unifiable algebra P the identity relation idP belongs to ConF (P), i.e., P embeds into F. The latter definition is new. One can think about it as an “exact” analog of projective unification. Fact 5.1. Let Q be a quasivariety. Then 1. Q has unitary (unitary exact) unification if and only if V(Q) has unitary (unitary exact) unification; 2. if V(Q) has projective (exact) unification then Q has projective (exact) unification; 3. the converses of the implications from (2) do not hold. Proof. Let R be a Q-finitely presented unifiable algebra and P be a V(Q)-finitely presented algebra, both defined by the same formula. Let p : P → R be the surjective homomorphism from Fact 2.4. (1) It follows from Fact 2.4 that the assignment u → u ◦ p maps bijectively the set of unifiers for R onto the set of unifiers of P. In particular, ker p is contained in every unifier of P and ConF (P) is isomorphic, as an ordered set, to ConF (R). This yields that R has unitary exact unification iff R has unitary exact unification. Note also that u is a most general unifier for R iff u ◦ p is a most general unifier for P. Actually, the equivalence holds for all unification types considered in [64]. (2) Now let us assume additionally that R is unifiable. Then the algebra P is also unifiable. Hence, by the assumption in the implication, there exists an injective unifier u : P → F. Again by Fact 2.4, there exists u ¯ : R → F such that u = u ¯ ◦ p. Thus p is injective and P and R are isomorphic. This shows that both implications from point (2) are valid. (3) The variety of (pointed) monounary algebras does not have exact unification, but its subquasivariety generated by free algebras has projective unification, see Examples 7.1 and 7.2. 2 Note that projective unification is the strongest property among four unification types considered here. Indeed, as projective algebras are reducts of free algebras, projective unification yields unitary unification and exact unification. Moreover, both these properties yield unitary exact unification. For ASC quasivarieties we have the following collapse.

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Proposition 5.2. An ASC quasivariety has unitary exact unification if and only if it has exact unification. In particular, ASC and unitary unification yield exact unification. Proof. The backward direction is obvious. For the forward direction let us consider a Q-finitely presented unifiable algebra P. By Corollary 3.4, P ∈ SP(F). This means that there are congruences αi ∈ ConF (P),  i ∈ I, such that i∈I αi is the identity relation idP on P . Since there is a least element in ConF (P), it must be idP . 2 Note however that SC together with unitary unification do not yield projective nor exact unification, see Example 7.1. Corollary 5.3. (Compare [26].) Every quasivariety with exact unification (in particular, with projective unification) is ASC. Proof. It follows directly from Theorem 3.1 point (4).

2

Recall that a quasivariety is hereditary ASC if all its subquasivarieties are ASC. Under an additional assumption projective unification yields also this property. Lemma 5.4. Let Q be a quasivariety with projective unification and R be a subquasivariety of Q. Assume that there exists an algebra C which embeds into some free algebra for Q and also embeds into some free algebra for R. Then R has projective unification. Proof. Let R be an R-finitely presented unifiable algebra. Let P be a Q-finitely presented algebra given by the same presentation (defining formula) as P. From the assumption it follows that an algebra in Q or in R is unifiable (in Q or in R, respectively) iff it admits a homomorphism into C (note that we adopt the convention that all considered algebras, including C, have nonempty carriers). Thus P is also unifiable, and hence projective in Q. We have to show that R is projective in R. To this end, let us consider a surjective homomorphism g : A → B between algebras from R and a homomorphism f : R → B. We show that f factors through g. By Fact 2.4, there is a surjective homomorphism p : P → R such that every homomorphism h : P → A, where A ∈ R, factors through p. Since P is projective in Q, there is a homomorphism f¯: P → A such that g ◦ f¯ = f ◦ p. It follows that there exists a homomorphism f¯¯: P → A such that f¯ = f¯¯ ◦ p. Thus we have f ◦ p = g ◦ f¯ ◦ p, and the surjectivity of p yields f = g ◦ f¯. The proof is illustrated by the following commutative diagram (surjective homomorphisms are denoted by two headed arrows). p

P f¯

A

R

f¯ f

g

2

B

Corollary 5.5. Let Q be a quasivariety with projective unification. Assume that there exists an algebra C such that for every nontrivial subquasivariety R of Q algebra C embeds into some free algebra for R. Then Q is hereditary ASC. Proof. It follows from Corollary 5.3, Lemma 5.4 and the fact that trivial quasivarieties are ASC. 2 The condition from Corollary 5.5 is a bit technical. But in the case when there is some constant in the language, it is equivalent to the transparent one.

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Corollary 5.6. Let Q be a quasivariety with projective unification and such that there exists a constant in the language of Q. Assume that for every nontrivial subquasivariety R of Q a free algebra of rank 0 for R is also free of rank 0 for Q. Then Q is hereditary ASC. Proof. Indeed, the assumption guarantees the existence of a free algebra of rank 0 for Q. By assumption, this algebra is also free in all nontrivial subquasivarieties of Q. Thus it may be taken as C in Corollary 5.5. 2 Remark 5.7. The “exact” analogue (i.e., when projective unification is replaced by exact unification) of Corollary 5.5 does not hold. This is witnessed by the variety of De Morgan lattices [65]. Indeed, it has three nontrivial subvarieties: the variety generated by a two-element De Morgan lattice, the variety of Kleene lattices, and itself. All of them have the same free algebras of rank one. They are isomorphic to the square of a two-element De Morgan lattice. One may deduce from [14] that the variety of De Morgan lattices has exact unification. However the variety of Kleene lattices is not ASC [65]. Remark 5.8. The property of having projective unification is not hereditary. Example 7.2 shows that the assumption on the algebra C in Corollary 5.5 and the assumption on a free algebra of rank 0 in Corollary 5.6 cannot be dropped. Verification of projective unification has in general a syntactical nature. However, having projective unification is a stronger property than having exact unification (see Theorem 8.12 or the variety of De Morgan lattices from Remark 5.7), and it is not surprising that sometimes it may be established more easily with the aid of semantical methods. We demonstrate this in Example 7.9. Corollary 3.2 yields that if Q is ASC and C  F, then every algebra of the form A × C belongs to Q(F), where A ∈ Q. On the other hand, if Q has projective unification, then every nontrivial Q-finitely presented algebra P in Q(F) is of the form B × C, where C  F, in a superficial way, i.e., with B trivial and C ∼ = P. Suppose that Q has projective unification and F has a minimal subalgebra C. Is it then true that every nontrivial (finitely generated or V-finitely presented or just finite) algebra A in Q(F) has C as a direct factor? In general: no. We demonstrate this in Example 7.10. Still, we have the following fact (see Example 7.9 for the definition of discriminator variety). Proposition 5.9. (See [1, Corollary 2.2].) Suppose V is a discriminator variety in a finite language. If C is a finite homomorphic image of a finitely generated member A of V, then C is a direct factor of A. Corollary 5.10. Let V be a discriminator variety in a finite language. Assume that there is a minimal finite subalgebra C of F (e.g., when F(0) exists and is finite). Then for every nontrivial V-finitely presented algebra P the following equivalence holds: P ∈ Q(F) if and only if P ∼ = B × C for some B ∈ V. Proof. By Corollary 5.3 and [12, Theorem 3.1], the backward implication holds. For the other direction take a nontrivial V-finitely presented algebra P in Q(F). Then P ∈ SPPU (F). Since P is V-finitely presented, analogically as in the proof of Corollary 3.4, we deduce that P ∈ SP(F). This yields that for every pair of distinct elements in P there is a homomorphism from P into F separating them. Since P is nontrivial, there exists at least one such homomorphism h : P → F. Clearly, F admits a homomorphism g onto C. Since C does not have proper subalgebra, g ◦ h maps P onto C. Thus, by Proposition 5.9, C is a direct factor of P. 2 6. Striving for finiteness In order to check the conditions from Theorem 3.1 it is possible that one has to work on infinite algebras. The following question arises: When can we simplify verification of ASC by restricting the condition (3) from

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Theorem 3.1 to finite algebras? In this section we propose a solution to this problem, namely Theorem 6.1. In the next section we will show some of its applications. Let us start by recalling necessary notions. We say that a class K of algebras has the finite model property (FMP for short) if V(K) is generated, as a variety, by finite members of K. Note that it may happen that a quasivariety does not have the FMP while the variety it generates does. A class K has the strong finite model property (SFMP for short) if Q(K) is generated, as a quasivariety, by finite members from K. In particular, every locally finite quasivariety (i.e., in which every finitely generated algebra is finite) has the SFMP. A quasivariety Q has equationally definable principal relative congruences (EDPRC for short and EDPC for varieties) if there exists a finite family of equations sk (u, v, x, y) ≈ tk (u, v, x, y), k  n, such that for every a, b, c, d ∈ A and A ∈ Q (c, d) ∈ θQ (a, b)

iff

A |=



sk (c, d, a, b) ≈ tk (c, d, a, b).

kn

Theorem 6.1. Let Q be a quasivariety in a finite language with the FMP and EDPRC. Assume that F has a finite Q-simple subalgebra C. Then Q is ASC if and only if for every finite Q-subdirectly irreducible algebra S we have SF

or

S × C  F.

Let us emphasize that the assumptions of Theorem 6.1 are very natural from the perspective of logic. Indeed, assume that a deductive system S is algebraizable and Q is its equivalent quasivariety. Then having EDPRC by Q corresponds to the deduction-detachment theorem for S [9, Theorem 5.5], [19, Theorem 4.6.13]. The algebra C may be often chosen as an algebra with elements which correspond to verum and falsum. Moreover, the FMP for K together with the existence of a finite axiomatization for V(K) is commonly used to show that the equational theory of K is decidable, and hence that Th(S) is decidable [58, Theorem 3]. Lemma 6.2. Assume that Q has the SFMP and C is a subalgebra of F. If for every finite algebra S ∈ QSI , S × C ∈ Q(F), then Q is ASC. Proof. By Proposition 4.1, the class of finite Q-subdirectly irreducible algebras is contained in ASCC(Q). Consequently, by Proposition 2.1, all finite algebras from Q are in ASCC(Q). Thus, by the SFMP, we obtain that Q = ASCC(Q). This means that Q is ASC. 2 For congruences α of A and β of B let α × β be the congruence of A × B given by {((a1 , b1 ), (a2 , b2 )) ∈ (A × B)2 | (a1 , a2 ) ∈ α and (b1 , b2 ) ∈ β}. A quasivariety Q has the Fraser–Horn property (FHP for short) if for all algebras A, B in Q every Q-congruence of the product A × B decomposes as α × β, where α is a Q-congruence of A and β is a Q-congruence of B. Every relative congruence distributive quasivariety has the FHP but this notion is more general, see [20]. Lemma 6.3. Assume that Q is a quasivariety in a finite language with the FHP and F has a finite Q-simple subalgebra C. If Q is ASC then for every finite S ∈ QSI we have SF

or

S × C  F.

Proof. Let S be a finite Q-subdirectly irreducible algebra. By ASC, S × C ∈ Q(F) = SPPU (F). This means that for each pair of distinct elements in S × C there is a homomorphism from S × C into an ultrapower of

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F that separates them. Let (a, b) ∈ S 2 be a pair of distinct elements which belongs to every Q-congruence of S except the equality relation idS on S. Further, let c be an element of C. Let h : S × C → G be a homomorphism such that h(a, c) = h(b, c), where G is an ultrapower of F. Then the FHP yields that ker(h) = α × β, where α is a Q-congruence of S and β is a Q-congruence of C. As h(a, c) = h(b, c), α equals idS and, by Q-simplicity of C, β equals idC or C 2 . Thus, either S or S × C embeds into G. By the finiteness of the language of Q and the finiteness of both algebras, at least one of them embeds into F. 2 We need two facts from the literature. Proposition 6.4. (See [10, Theorem 3.3].) For a quasivariety the FMP and EDPRC yield the SFMP. Proposition 6.5. (See [9, Theorem 5.5], [19, Theorem Q.9.3].) A quasivariety with EDPRC is relative congruence-distributive, and thus has the FHP. Proof of Theorem 6.1. For the backward direction combine Proposition 6.4 and Lemma 6.2. For the forward direction combine Proposition 6.5 and Lemma 6.3. 2 As a matter of fact, there is an analog of Theorem 6.1 for SC. Corollary 6.6. Let Q be a quasivariety in a finite language with EDPRC and the FMP. Then Q is SC if and only if every finite Q-subdirectly irreducible algebra is a subalgebra of F. Proof. The backward direction follows from Proposition 6.4 and the fact that all finite algebras from Q are in Q(F). This fact follows from Proposition 2.1 and the assumption. The forward implication may be proved similarly, but easier, as Lemma 6.3. 2 Remark 6.7. Corollary 6.6 was obtained in [78, Theorem 5.1.8] under an additional condition. However, in the cases of intermediate logics and of normal extensions of the modal logic K4 the formulation presented there is the same as ours [78, Corollary 5.1.10]. Remark 6.8. Having in mind potential applications, it is worth to stress that the characterizations of ASC and SC formulated in Theorem 6.1 and Corollary 6.6 are valid when the assumptions of the FMP and EDPRC are replaced by the SFMP and the FHP. In particular, it holds for locally finite relative congruencedistributive quasivarieties. Several forms of definability of relative principal congruences which are weakenings of EDPRC were considered in the literature. They correspond to variants of the deduction-detachment theorem for deductive systems. Among them the property of having equationally semi-definable principal relative congruences, corresponding to the contextual deduction-detachment theorem [73, Theorem 9.2], is sufficient for Theorem 6.1 to work. Indeed, having equationally semi-definable principal relative congruences yields relative congruence-distributivity, and with the FMP yields the SFMP [73, Theorem, 8.7, Corollary 3.7]. Problem 6.9. Is it possible to weaken the assumption of Theorem 6.1 of having EDRPC to having the relative congruence extension property, corresponding to the local deduction-detachment theorem [8, Corollary 3.7], or to having parameterized equationally definable principal relative congruences, corresponding to the parameterized deduction-detachment theorem [19, Section 2.4]?

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7. Examples In this section we give several examples of ASC varieties. The main objective is to present varieties which characterize ASC\SC logics. Exception are varieties of (pointed) monounary algebras and varieties of bounded lattices. Our examples are intended to illustrate how Theorems 3.1 and 6.1 and the techniques used in their proofs may be applied. Also the example of monounary algebras shows that there does not have to exist a largest ASC subquasivariety of a given quasivariety. The example of pointed monounary algebras shows that the property of having projective unification is not hereditary. Moreover, the example of bounded lattices shows that a certain “plausible” condition for ASC is actually strictly weaker than ASC. We will use a nonstandard notation for operations in algebras and instead of ∨, ∧, →, ¬ we will write ∨ ∨, ∧ ∧, ⇒, ¬ symbols. We do so in order to make a visible distinction between the language and the metalanguage. Example 7.1 (Monounary algebras). Let V be the class of all monounary algebras. These are algebras with just one basic operation which is unary. It is denoted by f . We claim that ASCC(V) is defined by the quasi-identity j = (∀x, y)[f (x) ≈ f (y) → x ≈ y]. We identify a free monounary algebra F(1) with (N, f : x → x + 1). Note that F is isomorphic to a disjoint union of denumerably many copies of F(1). We clearly have F |= j and F |= j ∗ . Hence ASCC(V) |= j. Now in order to prove our claim it is enough to show that the quasivariety defined by j is ASC. To this end one may use the condition (2 ) from Corollary 3.2. Indeed, if A |= j, then A × F(1) is a disjoint union of the subalgebras generated by (a, n), a ∈ A, n ∈ N, where a ∈ / f (A) or n = 0. Every one of these subalgebras is isomorphic to F(1). Therefore A × F(1) is free for V and belongs to Q(F). Let us consider the variety W defined by (∀x, y)[f 2 (x) ≈ f 2 (y)]. It is generated by a 3-element subdirectly irreducible algebra R, where R = {a, b, c} and f (a) = b, f (b) = f (c) = c. The variety W has, up to isomorphism, two subdirectly irreducible algebras: R and its subalgebra S with the carrier S = {b, c}. The structure of free algebras for W is transparent too. For a cardinal number κ a free algebra of rank κ for W is isomorphic to the quotient of κ copies of R where all copies of c are glued together. It follows that W is SC. However j is not valid in W and W  ASCC(V). This shows that there does not need to exist a largest (A)SC subquasivariety of a given quasivariety. (For this single purpose a simpler variety, defined by (∀x, y)[f (x) ≈ f (y)], would be considered.) The variety W does not have exact (nor projective) unification. Since W is finitely generated, the class of W-finitely presented algebras coincides with the class of finite members of W. Clearly, every such algebra is unifiable. Let R be the extension of R given by R = R ∪ {a }, where a ∈ / R and f (a ) = b. Then R is not a subalgebra of any free algebra for W. Note that S is isomorphic to a subalgebra of a free algebra for W, but it is not projective. Still, W has unitary unification. In order to see this let us consider a nontrivial finite algebra P ∈ W. We have P = f −2 (r), where r = f 2 (p) for every p ∈ P . Since P is not trivial, the set f −1 (r) − {r} = {p0 , . . . , pk−1 } is not empty. Now a most general unifier u for P may be defined by putting ⎧ ⎪ ⎪ ⎨vi u(p) = f (vi ) ⎪ ⎪ ⎩f 2 (v ) 0

if p ∈ f −2 (r) − f −1 (r) and f (p) = pi if p ∈ f −1 (r) − {r} and p = pi if p = r

where v0 , . . . , vk−1 are free generators in a free algebra for W.

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Example 7.2 (Pointed monounary algebras). Let us extend the language from Example 7.1 by a constant c. We call algebras in this language pointed monounary algebras. Let Q be a quasivariety of such algebras defined by the quasi-identity j from the previous example. Then Q has projective unification (actually, this is also true for ASCC(V) in Example 7.1). Indeed, an algebra A = (A, f, c) in Q admits a homomorphism into F iff every one-generated subalgebra of (A, f ) is isomorphic to (N, f : x → x+1), the algebra (Z, f : x → x+1) does not embed into (A, f ), and c ∈ / f (A). This is equivalent to the freeness of A for Q (and for the variety of all pointed monounary algebras). Let R be a subquasivariety defined by the identity f (c) ≈ c. Then FR (0), a free algebra of rank 0 for R, is trivial. Let A be an algebra where A = {a, c} and f (a) = a, f (c) = c. Then A × FR (0) ∼ = A and A |= (∀x)[f (x) ≈ x → x ≈ c]. But all free algebras for R satisfy this quasi-identity. Thus, by Corollary 3.2, R is not ASC. Also, by Corollary 5.3, R does not have projective unification. This shows that the property of having projective unification is not hereditary. Notice also that it is not difficult to modify the above example in order to obtain a varietal one. Indeed, instead of Q let us consider the variety V of pointed monounary algebras defined by the identity (∀x)f 2 (x) = x. Instead of R let us take the subvariety W of V defined by f (c) ≈ c. Then V has projective unification and the same algebra A witnesses that W is not ASC. Example 7.3 (Varieties of bounded lattices). By a bounded lattice we mean an algebra L with a lattice reduct and with two constants 0 and 1 which are the bottom and the top elements in L respectively. Due to the lack of the FMP, Theorem 6.1 does not apply to all varieties of bounded lattices. (For instance, the variety defined by the modularity law does not have the FMP. In [32] an identity e was found that holds in all finite modular lattices, but does not hold is some infinite one L. Clearly, e holds in all finite bounded modular lattices as well but it does not hold in the bounded expansion of L.) Still, the argument from the proof may be used to show that there are only two ASC (SC in fact) varieties of bounded lattices. Strictly, the proof of Lemma 6.3 yields also the following fact. Lemma 7.4. Assume that Q is a quasivariety with the FHP and that F has a finite Q-simple subalgebra C. If Q is ASC then for every S ∈ QSI there exists an ultrapower G of F such that SG

or

S × C  G.

Here by 2 we denote the bounded lattice ({0, 1}, ∧ ∧, ∨ ∨, 0, 1). Note that 2 is free of rank zero for every nontrivial variety of bounded lattices. The following two lemmas are folklore. Lemma 7.5. Let V be a nontrivial variety of bounded lattices and G be an ultrapower of F. Then 1 is join-irreducible in G. Proof. Since the conclusion of the lemma is expressible by a first order sentence and G is elementarily equivalent to F, it is enough to prove it for F. v), q(¯ v ) < 1 and Assume that 1 is join-reducible in F. Then there are terms p, q such that such that p(¯ p(¯ v )∨ ∨q(¯ v ) = 1 in F. This yields that V satisfies the identity pˆ∨ ∨ˆ q ≈ 1, where pˆ = p(0, . . . , 0), qˆ = q(0, . . . , 0). In particular, in F(0), which is isomorphic to 2, we have pˆ∨ ∨ˆ q = 1. Thus at least one of pˆ, qˆ, say pˆ, equals 1 in F(0) and in F. Since in bounded lattices all term operations are monotone, pˆ  p. Hence p = 1 in F. This gives a contradiction. 2 Let N5 be a 5-element lattice in which the non-top and the non-bottom elements form a disjoint union of a singleton with a two-element chain (exactly two among these elements are comparable). Let M3 be a 5-element lattice in which the non-top and the non-bottom elements form a three-element antichain (all

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of them are incomparable). Let Nb5 and Mb3 be the bounded lattices with the lattice reducts N5 and M3 respectively. By the distributive law we mean the identity (∀x, y, z) [x∧ ∧(y∨ ∨z) ≈ (x∧ ∧y)∨ ∨(x∧ ∧z)]. Lemma 7.6. Let V be a variety of bounded lattices. Then the following conditions are equivalent: 1. V satisfies the distributive law, 2. V = V(2) or V is the trivial variety, 3. Nb5 ∈ / V and Mb3 ∈ / V. Proof. (1)⇒(2) It follows from the fact that 2 is the only, up to isomorphism, subdirectly irreducible bounded distributive lattice (one may deduce it from e.g. [21, Theorem 10.21]). (2)⇒(1)⇒(3) It is routine. (3)⇒(1) Assume that in V there is a non-distributive bounded lattice L. Then its lattice reduct has a sublattice K isomorphic to M3 or N5 [21, Theorem I4.10]. Let Kb be a bounded sublattice of L generated by K. Note that the lattice reduct of Kb may differ from K only by having an additional element on the top and/or having an additional element in the bottom. In any case, Kb has one of bounded lattices Mb3 , Nb5 as a homomorphic image. Thus one of these algebras belongs to V. 2 Proposition 7.7. Let V be a variety of bounded lattices. Then the following conditions are equivalent: 1. V is SC, 2. V is ASC, 3. V satisfies the distributive law. Proof. (1)⇒(2) It is obvious. (2)⇒(3) Assume that in V the distributive law does not hold. Then, by Lemma 7.6, at least one of Mb3 , Nb5 belongs to V. For convenience, let us denote it by S. Clearly, S is finite, subdirectly irreducible and has the top element join-reducible. Since in S × 2 the top element is also join-reducible, Lemma 7.5 yields that neither S nor S × 2 embeds into any ultrapower of F. Thus, by Lemma 7.4, V cannot be ASC. (3)⇒(1) By Lemma 7.6, there are only two varieties of distributive bounded lattices: the trivial one, which is clearly SC, and the minimal one V(2). In fact V(2) is minimal as a quasivariety as well and, as such, is SC. 2 We finish this example by making one remark. Consider the condition obtained by combining the conditions from Theorem 3.1. (6) For every S ∈ QSI , S → F yields S ∈ Q(F). Theorem 3.1 shows that (6) follows from ASC. But is not equivalent to ASC. In order to see this, let us consider the variety V generated by Mb3 . By Proposition 7.7, V is not ASC. Let us check that the condition (6) is fulfilled. There are, up to isomorphism, exactly two subdirectly irreducible algebras in V: 2 and Mb3 . Clearly 2 ∈ Q(F). Moreover, Mb3 does not admit a homomorphism into F. Indeed, since Mb3 is simple, a homomorphic image of Mb3 would have just one element or be isomorphic to Mb3 . The first case is impossible since F does not have idempotents. The second case is also impossible as we showed in the proof of Proposition 7.7. Note that the unbounded case is different. In particular, V(M3 ) is SC [47].

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Let us move to examples that come from logic. We are mainly interested in normal modal logics, and, in particular, in normal extensions of the transitive and reflexive modal logic S4. Every such extension has an adequate semantics given by a variety of closure algebras [60], [22, Chapter 10]. An algebra M is a modal algebra if it has a Boolean algebra reduct and besides Boolean operations it has a unary operation 3 such that for all a, b ∈ M 30 = 0

and

3(a∨ ∨b) = 3a∨ ∨3b.

If in addition for every a ∈ M the algebra M satisfies a  3a = 33a ¬x. An element a of a closure algebra is closed (open) if a = 3a we call it a closure algebra. Let 2x = ¬3¬ (a = 2a respectively). We picture a closure algebra M by drawing the Hasse diagram of the ordered set (M, ), where  is given by the lattice structure of M. We draw closed elements as 3, open as 2, open and 3 , and others as ◦. The simplest nontrivial closure algebra, denoted by 2 and depicted in Fig. 1, closed as 2 has two elements and 3 operation acts on it identically. In particular it is term-equivalent to a two-element Boolean algebra. It is important to note that 2 embeds into every nontrivial closure algebra. Moreover, 2 is free of rank zero for every nontrivial variety of closure algebras. Let us recall that congruences of a closure algebra M are in one to one correspondence with open filters of M, i.e., Boolean filters which are additionally closed under 2 operation. For a congruence α its corresponding open filter is the class 1/α. It follows that the variety of closure algebras is congruence-distributive. Actually, a stronger statement is true: every variety of closure algebras has EDPC witnessed by the equation 2(x ⇔ y) ⇒ (u ⇔ v) ≈ 1. Note also that each element a of a closure algebra M which is open and closed yields a direct product decomposition M ∼ = M/α × M/β, where α is a congruence generated by (1, a) and β is a congruence generated by (1, ¬a). Example 7.8 (Varieties of monadic algebras). A closure algebra M is a monadic algebra if for all a ∈ M we have 32a = 2a. This means that all open elements in M are also closed. Recall that varieties of monadic algebras form adequate semantics for normal extensions of the transitive, reflexive and symmetric modal logic S5 [60], [22, Chapter 10]. As we already noted in the introduction, every variety of monadic algebras is a discriminator variety. Hence it has projective unification and is ASC. Since the variety of monadic algebras is a prototypical example of an ASC variety which is not SC, let us look at it from an algebraic perspective. For this purpose we recall basic facts about monadic algebras. For a positive integer l let Sl be a closure algebra with l atoms and with 0 and 1 as the only closed elements. The algebra S1 , which is isomorphic to 2, and the algebra S2 are depicted in Fig. 1. Clearly, all Sl are monadic. Let V be a variety of monadic algebras. Then V is semisimple, i.e., all its subdirectly irreducible algebras are simple. Moreover, every finite simple closure algebra is isomorphic to one of Sl [38, Lemma 8, Theorem 7], [51, Theorem 4.2]. This gives that every finite monadic algebra M is isomorphic to a product of those Sl which are its homomorphic images. Indeed, every maximal congruence α of a monadic algebra M is generated by a pair (a, 1) where a is open, and hence also closed, in M. Thus M is isomorphic ¬a, 1). Since V is locally finite [2], this applies to to the product M/α × M/β, where β is generated by (¬ F(k) for every finite k. So we have F(k) ∼ =

m

l=1

Sdl l

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for some natural numbers m, d1 , . . . , dm . Note also that if Sl ∈ V and k  l, then Sl is a homomorphic image of F(k) and dl  1. (The exact structure of FW (k) may be deduced from [2,39,51] where free monadic algebras are described.) Let us use Theorem 6.1 in order to show that V is ASC. As we already noted, V has EDPC and, since it is locally finite, it has the FMP. Moreover a two-element closure algebra 2 embeds into every nontrivial monadic algebra. Thus the assumptions of Theorem 6.1 hold. Let us verify the condition from this theorem. If V is trivial, it vacuously holds. So assume that Sl ∈ V for some positive integer l. Take n  l. Then F(n) ∼ = Sl × M for some nontrivial monadic algebra M, and hence Sl × 2 embeds into F(n). Note that there are only two SC varieties of monadic algebras, namely the trivial one and V(2) = SP(2). The latter one is actually term-equivalent to the variety of Boolean algebras. Indeed, all other varieties of monadic algebras contain S2 . Thus, as we indicated in the introduction and will prove in Proposition 8.7, they cannot be SC. Since, V is a discriminator variety, it also follows from [15, Corollary 5.6]. Example 7.9 (Locally finite discriminator varieties). Actually, the argument for ASC from the previous example may be used in a more general setting. Recall that a variety V is a discriminator variety if it is generated by a class K of algebras for which there is a term t(x, y, z) such that for all a, b, c ∈ A, A ∈ K we have  t(a, b, c) =

a

if a = b

c

if a = b

.

Assume that V is a locally finite discriminator variety in a finite language and that there exists an algebra C that embeds into every nontrivial member of V. Then V has the FMP and EDPC [6, Page 200]. These assumptions are met in e.g. the varieties of monadic algebras [51], of MVn -algebras [17], in locally finite varieties of relation algebras [56] or diagonal free cylindric algebras [40]. By Proposition 5.9, every subdirectly irreducible (which is here the same as simple) algebra S in V is a direct factor of F(n) for n  |S|. Thus if S is not isomorphic to F(n), then it is a proper direct factor of F(n) and then S × C embeds into F(n). Thus the assumptions and the condition form Theorem 6.1 hold and V is ASC. Let us move to more complicated examples, varieties of closure algebras which are not discriminator varieties. Example 7.10 (Varieties of S4.3-algebras). Let VS4.3 be the variety generated by the closure algebras in which open elements form a chain. Alternatively one may define VS4.3 , relative to the variety of closure algebras, by (∀x, y)[2(2x ⇒ y)∨ ∨2(2y ⇒ x)]. Note that VS4.3 characterizes the modal logic S4.3, see e.g. [16]. Let V be a subvariety of VS4.3 . We already noted in the introduction that V is ASC. Let us now argue for it without projective unification. By Bull’s theorem [11], V has the FMP. Thus the assumptions of Theorem 6.1 are satisfied for V. Moreover, the condition from the theorem is verified in [77, Lemma 2]. Thus V is ASC. Note that Rybakov in [77, Theorem 5] obtained a quasi-equational basis for Q(F). In Section 5 we asked whether all finite/finitely presented/finitely generated nontrivial algebras in Q(F) have a minimal subalgebra of F as a direct factor provided V is a variety with projective unification. To falsify it let us consider the closure algebra M depicted in Fig. 2 and the subvariety V of VS4.3 containing M. Then M has 2 as a homomorphic image. Thus, by Theorem 3.1 point (4), M ∈ Q(F). Nevertheless, 2 is not a direct factor of M.

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Fig. 2. The algebra M from VS4.3 .

Now we move to varieties of Heyting algebras. A Heyting algebra (called sometimes a pseudo-Boolean algebra) H is a bounded lattice expanded by one binary operation ⇒ such that for all a, b, c ∈ H a∧ ∧c  b

iff

c  a ⇒ b.

Let ¬x = x ⇒ 0. Varieties of Heyting algebras constitute an adequate semantics for intermediate logics. In particular, the class of all Heyting algebras, which turns out to be a variety, characterizes intuitionistic logic [16, Chapter 7]. As in the case of closure algebras, there is exactly one minimal (quasi)variety of Heyting algebras. It is generated by a two-element Heyting algebras, again denoted as 2. Corollary 3.5 and Fact 7.12 yield that for varieties of Heyting algebras SC is in fact equivalent to ASC. Nevertheless, they are strongly connected to varieties of closure algebras. In the next section we will show how to construct, starting from an SC variety of Heyting algebras, infinitely many varieties of closure algebras which are ASC\SC. Example 7.11 (Levin and Medvedev varieties). Recall that with every ordered set O, the Heyting algebra O+ of its up-sets is associated. Then O, treated as an intuitionistic frame, validates the intuitionistic formula t(¯ x) iff O+ validates the identity (∀¯ x)[t(¯ x) ≈ 1], see e.g. [16, Chapter 7]. For a natural number n let (2, )n be the power of the ordered set (2, ) with 2 = {0, 1} and 0  1. Let Levn be the ordered set obtained from (2, )n by removing the top element. Since the algebra Lev+ 2 , depicted in Fig. 3, is important in our 2 investigations, we use a more intuitive notation 2 ⊕ 1 for it. Note that the logic characterized by all Levn is Medvedev finite problems logic [61,62], and an intermediate logic is one of Levin’s logics [54] iff it is Medvedev logic or it is characterized by one of the frames Levn [80, Theorem 3.1].

Fig. 3. The Heyting algebra 22 ⊕ 1.

For n ∈ N let VLevn = V(Lev+ n) and VMed be their varietal join VMed =

 n∈N

VLevn .

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We need the following basic property of Heyting algebras. It may be deduced from e.g. [75, Statement VI.6.5]. Fact 7.12. Let H be a Heyting algebra, a ∈ H and a = 0. Then there is a homomorphism h : H → 2 such that h(a) = 1. Lemma 7.13. Let V be a nontrivial variety of Heyting algebras. Then 22 is a V-finitely presented unifiable algebra. Proof. Since the language of V is finite and 22 is a finite algebra belonging to V, Fact 2.5 yields that 22 is V-finitely presented. It may be shown that 22 is isomorphic to a finitely presented Heyting algebra defined ¬x ≈ 1. by the formula x∨ ∨¬ The existence of a projection from 22 onto 2 yields that 22 is unifiable. Note that 2 is free for every nontrivial variety of Heyting algebras. 2 Lemma 7.14. Let V be a variety of Heyting algebras containing 22 ⊕ 1. Then 22 does not embed into F. Proof. Striving for a contradiction, suppose that 22 embeds into F. Then there is a term t such that in F we have ¬t = 1, t∧ ¬t = 0. 0 < t < 1, 0 < ¬t < 1, t∨ ∨¬ ∧¬ Note that neither t nor ¬t is a Boolean tautology. Indeed, by Fact 7.12, there is a homomorphism k : F → 22 ¬t) = (0, 1). The Heyting algebra 22 ⊕ 1 has two atoms a = {(1, 0)}, such that k(t) = (1, 0) and k(¬ ¬a = {(0, 1), (1, 0)}. Let g : V → 22 ⊕ 1 be the mapping given by ∨¬ ¬a = {(0, 1)}, and one coatom a∨ ⎧ ⎪ a ⎪ ⎪ ⎪ ⎨¬a g(v) = ⎪ ¬a a∨ ∨¬ ⎪ ⎪ ⎪ ⎩ 0

if k(v) = (1, 0) if k(v) = (0, 1) if k(v) = (1, 1)

,

if k(v) = (0, 0)

and g¯ : F → 22 ⊕ 1 be the homomorphic extension of g. Let h : 22 ⊕ 1 → 22 be a surjective homomorphism ¬a onto 1 and a onto (1, 0). Note that h−1 (1) = {1, a∨ ¬a} is the only coset containing more that maps a∨ ∨¬ ∨¬ than 1 element. We have k|V = h ◦ g and hence, by the universal mapping property for F, k = h ◦ g¯. ¬t) = ¬a. Now we compute in 22 ⊕ 1 Therefore g¯(t) = a and g¯(¬ ¬t) = g¯(t)∨ ¬g¯(t) = a∨ ¬a < 1. 1 = g¯(1) = g¯(t∨ ∨¬ ∨¬ ∨¬ This leads to a contradiction. 2 Note that if V is a variety of Heyting algebras containing a three-element algebra, then 22 is not projective in V, see the last remark in [35]. Proposition 7.15. Let V be one of the varieties VMed , VLevn for n  2. Then 1. V is SC, 2. V does not have unitary exact (and hence unitary, neither exact nor projective) unification.

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Proof. (1) For VMed it was proved by Prucnal [71]. For VLevn a small modification of Prucnal’s proof works [80, Lemma 3.2]. (2) It follows from Lemmas 7.13 and 7.14 and Proposition 5.2. 2 The lack of unitary unification in Proposition 7.15 follows also from [35, Theorem 4.4] and from [23, Lemma 3]. It was shown there that every variety of Heyting algebras containing 22 ⊕ 1 cannot have unitary unification even without assuming SC. Indeed, then the algebra 22 does not have a most general unifier. In the next section, based on Example 7.11, we will construct ASC\SC varieties of closure algebras for which points (2) and (3) in Proposition 7.15 will be also valid. 8. ASC for varieties of closure algebras and normal modal logics In this section we show that the varietal join V = U ∨ W of an SC variety of closure algebras U and a non-minimal nontrivial variety of monadic algebras W is ASC\SC. By Corollary 3.4, for an ASC variety V every V-finitely presented unifiable algebra is isomorphic to a subalgebra of a power of F. Therefore a finite unifiable algebra is a subalgebra of a power of some F(k). However in every known example of an ASC\SC variety V of modal algebras all V-finitely presented unifiable algebras actually embed into F. We claim that this fact is connected to the limitation of techniques used so far, not to any intrinsic property of modal algebras. We have already presented in Proposition 7.15 varieties of Heyting algebras, which are SC, with a finitely presented algebra not embeddable into any free algebra. We prove in Theorem 8.12 that if V is as in the first sentence of this section and additionally U contains the algebra B(22 ⊕ 1) shown in Fig. 5, then there is a finite, and therefore V-finitely presented, unifiable algebra in V which is not embeddable into F. This means that V does not have exact unification. Thus V does not have unitary exact (and hence neither unitary nor projective) unification. For such U we can take minimal modal companions of Levin varieties from Example 7.11. To the best of our knowledge, these are the first found examples of ASC\SC varieties of modal algebras without projective unification. 8.1. Join of varieties of McKinsey algebras and of monadic algebras Let μ(x) = 23x ⇒ 32x be the McKinsey term. A modal algebra M is a McKinsey algebra if it satisfies the McKinsey identity (∀x) μ(x) ≈ 1. McKinsey algebras have appeared in our investigations due to the fact that an ASC variety of closure algebras is SC iff it satisfies the McKinsey identity. Moreover, finitely generated algebras from the varietal join of a variety of McKinsey algebras and of a variety of monadic algebras are products of McKinsey algebras and monadic algebras. These facts are used to verify that a varietal join of an SC variety of closure algebras with a non-minimal nontrivial variety of monadic algebras is ASC\SC. Now we present their proofs. In what follows U denotes a variety of McKinsey algebras, W denotes a variety of monadic algebras, and V = U ∨ W = V(U ∪ W) denotes their varietal join. We add subscripts in the notation of free algebras denoting varieties for which these algebras are free. For instance a free algebra for V of rank ℵ0 , previously denoted by F, now is denoted by FV . Moreover, in this section 2 again denotes a two-element closure algebra.

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Recall from Example 7.8 that we may put FW (k) = 2d ×

m

Rl ,

l=1

where all Rl are not necessarily distinct finite simple monadic algebras with more than two elements, which means that they are in {Sl | l ∈ {2, 3, . . .}}, and d, m are some natural numbers. Let w1 , . . . , wk be free generators of FW (k). Let us interpret them as mappings with the domain {0, . . . , m} and w(0) ∈ 2d , w(l) ∈ Rl for l ∈ {1, . . . , m}. Define GW (k) =

m

Rl .

l=1

What we need to know about free generators in FW (k) is just the following fact. Lemma 8.1. For every index l ∈ {1, . . . , m} there exists i ∈ {1, . . . , k} such that wi (l) ∈ / {0, 1}. Proof. This holds since otherwise the subalgebra of FW (k) generated by w1 , . . . , wk would be a subalgebra l−1 m of 2d × j=1 Rj × 2 × j=l+1 Rj . This would contradict the fact that free generators generate the whole algebra FW (k). 2 Proposition 8.2. Let U be a nontrivial variety of McKinsey algebras, W be a variety of monadic algebras, and V = U ∨ W be their varietal join. Then FV (k) ∼ = FU (k) × GW (k). Proof. Let u1 , . . . , uk be free generators of FU (k) and w1 , . . . , wk be free generators of FW (k) interpreted as above. For i ∈ {1, . . . , k} let vi = (ui , wi ), where wi = wi |{1,...,m} . We prove that v1 , . . . , vk are free generators in FU (k) × GW (k) for the variety V. The verification of the universal mapping property may be split into the verification of two claims. Claim. The elements v1 , . . . , vk generate FU (k) × GW (k). Proof. The elements u1 , . . . , uk generate FU (k) and w1 , . . . , wk generate GW (k). Thus every element from FU (k) × GW (k) is of the form (s(¯ u), t(w ¯  )) for some terms s(¯ x), t(¯ x). Our aim is to find a term r(¯ x) such  that (s(¯ u), t(w ¯ )) = r(¯ v ). Define a term

m(¯ x) =

k  

μ(xi ).

i=1

Since U satisfies the McKinsey identity, in FU (k) we have m(¯ u) = 1. Let us compute m(w ¯  ) in GW (k). A routine verification shows that if a is neither a top nor a bottom element in Rl , then μ(a) = 0. Thus, by Lemma 8.1, for every l ∈ {1, . . . , m} there is i such that in Rl we have μ(wi (l)) = 0. Hence in GW (k) we ¬m(¯ have m(w ¯  ) = 0. Now for r(¯ x) = (m(¯ x)∧ ∧s(¯ x))∨ ∨(¬ x)∧ ∧t(¯ x)) we compute ¬m(¯ ¬m(w r(¯ v ) = ((m(¯ u)∧ ∧s(¯ u))∨ ∨(¬ u)∧ ∧t(¯ u), (m(w ¯  )∧ ∧s(w ¯  ))∨ ∨(¬ ¯  )∧ ∧t(w ¯  )) ∨(1∧ ∧t(w ¯  )) = (s(¯ u), t(w ¯  )). = ((1∧ ∧s(¯ u))∨ ∨(0∧ ∧t(¯ u), (0∧ ∧s(w ¯  ))∨

2

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Claim. For every algebra M ∈ V and every mapping f : {v1 , . . . , vk } → M there is a homomorphism f¯: FU (k) × GW (k) → M extending f . Proof. First observe that we do not have to verify the assertion for all M ∈ V. It is enough to show it for generators of V, see e.g. [53, Proposition 4.8.9]. We do it for algebras from U ∪ W. The case when M ∈ U is easy. Then for f¯ we just take the composition of the first projection of FU (k) × GW (k) with the homomorphism from FU (k) into M extending the mapping given by ui → f (vi ) for i ∈ {1, . . . , k}. Let us move to the case when M ∈ W. Since we assumed that U is nontrivial, 2d ∈ U. (Actually, 2d is free for V(2) of rank k and d = 2k .) Thus there is a homomorphism g : FU (k) → 2d such that g(ui ) = wi (0) for i ∈ {1, . . . , k} (recall that wi are free generators in FW (k)). Let g  : FU (k) × GW (k) → FW (k); (s, t) → (g(s), t) and h : FW (k) → M be a homomorphism such that h(wi ) = f (vi ) for i ∈ {1, . . . , k}. Then h ◦ g  : FU (k) × GW (k) → M is a homomorphism extending f . 2 This ends the proof of the proposition. 2 As a by-product, we obtain the following fact. Corollary 8.3. Let U be a variety of McKinsey algebras, W be a variety of monadic algebras, and V = U ∨ W be their varietal join. Then every finitely generated algebra in V is isomorphic to a direct product of algebras from U and W. Proof. The statement is obvious when U is trivial. So let us suppose that it is not the case. Let A ∈ V be k-generated. Then, by Proposition 8.2, there exists a congruence γ of FU (k) × GW (k) such that A is isomorphic to FU (k) × GW (k)/γ. Since V has the FHP, there are congruences α of FU (k) and β of GW (k) such that γ = α × β. Hence A∼ = FU (k)/α × GW (k)/β. Clearly A ∼ = FU (k)/α ∈ U and GW (k)/β ∈ W.

2

In [27] it was proved that a variety of S4.3-algebras is SC iff it satisfies the McKinsey identity. However the proof presented there actually uses only the fact that a variety of closure algebras under consideration is ASC. Let us formulate the reasoning in algebraic terms. We use the following lemma several times. Lemma 8.4. For every closure McKinsey algebra M and every open element a ∈ M which does not equal 0 there is a homomorphism h : M → 2 such that h(a) = 1. / 1/θ, and therefore θ < M 2 . Proof. Let θ be a congruence of M generated by (a, 1). Since a is open, 0 ∈  By the Zorn lemma, θ can be extended to a maximal congruence θ properly contained in M 2 . Then N = M/θ is simple. For closure algebras the property of being simple is equivalent to having exactly two open elements 0 and 1. The computation shows that in N we have μ(c) = 0 for every element c ∈ N − {0, 1}. Thus the inequality N = {0, 1} would contradict the satisfaction of the McKinsey identity by N (see also Proposition 8.5). Therefore N ∼ = 2. 2

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Recall that the four-element simple closure algebras, depicted in Fig. 1, was denoted by S2 . Note that V(S2 ) and the variety of McKinsey algebras is a splitting pair for the lattice of varieties of closure algebras. Proposition 8.5. (See [4, Example III.3.9], [76, Example IV.5.4].) Let U be a variety of closure algebras. Then S2 ∈ / U if and only if U satisfies the McKinsey identity. Lemma 8.6. Every SC variety of closure algebras satisfies the McKinsey identity. Proof. Suppose that U does not satisfy the McKinsey identity. Then, by Proposition 8.5, S2 ∈ U. Let ¬x ≈ 1 → 0 ≈ 1]. q = (∀x)[3x∧ ∧3¬ We have S2 |= q, and hence U |= q. But q is V(2)-passive. Since FV(2) is a homomorphic image of FU , the quasi-identity q is also U-passive, and therefore it holds in FU . Thus U is not SC. 2 Proposition 8.7. Let U be an ASC variety of closure algebras. Then the following conditions are equivalent: 1. U is SC, 2. U satisfies the McKinsey identity, 3. S2 ∈ / U. Proof. The equivalence (2)⇔(3) follows from Proposition 8.5, the implication (1)⇒(2) is given by Lemma 8.6 and the implication (2)⇒(1) follows from Corollary 3.5, and Lemma 8.4. 2 Now we may proceed to the main result of this section. Theorem 8.8. Let U be an SC variety of closure algebras and W be a non-minimal nontrivial variety of monadic algebras. Then the varietal join U = V ∨ W is ASC \SC. Proof. In the case when U is trivial we have V = W and the statement of the theorem was verified in Example 7.8. So we assume that U is nontrivial. Let us start by proving the following fact. Claim. The algebra FU (k) embeds into FV (k) for every natural number k. In particular, Q(FU ) ⊆ Q(FV ). Proof. By Lemma 8.4, there is a homomorphism h : FU (k) → GW (k) with the image isomorphic to 2. Then the homomorphism g : FU (k) → FU (k) × GW (k); t → (t, h(t)) embeds FU (k) into FU (k) × GW (k). By Propositions 8.2 and 8.7, the latter algebra is isomorphic to FV (k). 2 In order to verify ASC for V let us check the condition (3 ) from Corollary 3.2. Since V is congruencedistributive, every subdirectly irreducible algebra S from V is in U ∪ W [50, Corollary 4.2]. Case when S ∈ U. By the assumption that U is SC, S ∈ Q(FU ). Hence S × 2 ∈ Q(FU ), and by Claim, S × 2 ∈ Q(FV ). Case when S ∈ W. As explained in Example 7.8, for large enough k, S is a proper direct factor of FW (k). Thus, by Proposition 8.2, S is a proper direct factor of FV (k), and hence S × 2 embeds into FV (k). Finally note that, since W is non-minimal nontrivial, S2 ∈ V. Thus by Proposition 8.7, V is not SC. 2

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8.2. ASC without projective unification; modal companions of Levin and Medvedev varieties Let us briefly review the translation of intuitionistic logics into transitive reflexive normal modal logics in algebraic terms. Open elements of a closure algebra M form the Heyting algebra O(M) with the order inherited from M. Moreover, if W is a variety of closure algebras, then the class O(W) = {O(M) | M ∈ W} is a variety of Heyting algebras. The following fact was proved in [59, Section 1], see also [4, Chapter 1] and [5, Theorem 2.2]. Proposition 8.9. For every Heyting algebra H there is a closure algebra B(H) such that 1. OB(H) = H; 2. for every closure algebra M, if H  O(M), then B(H) is isomorphic to the subalgebra of M generated by H, The algebra B(H) is called the free Boolean extension of H. For each variety Y of Heyting algebras there is a variety of closure algebras Y  such that O(Y  ) = Y. Every such Y  is called a modal companion of Y. In general, there are many modal companions for a given variety of Heyting algebras. For instance the variety of monadic algebras is the greatest modal companion, and the variety V(2) is the smallest modal companion of the variety of Boolean algebras treated as Heyting algebras. The smallest modal companion of a variety Y of Heyting algebras is given by σ(Y) = V{B(H) | H ∈ Y} The importance of σ and O class operators follows from the Blok–Esakia theorem. It states that they are mutually inverse lattice isomorphisms between the subvariety lattice of the variety of Heyting algebras and the subvariety lattice of the variety of Grzegorczyk algebras [4, Theorem III.7.10], [16, Theorem 9.66], [85], [29]. Similarly to Example 7.11, we would like to find a V-finitely presented unifiable algebra that does not embed into FV . But this time we cannot take 22 . Indeed, Proposition 8.2 shows that in fact this algebra embeds into FV . But just a bit more complicated algebra does not embed. Let 4 be a four-element closure algebra depicted in Fig. 4. Note that V(4) and the variety of monadic algebras is a splitting pair for the lattice of varieties of closure algebras [5, Theorem 5.5]. We do not need this fact in full strength, but only the observation that 4 embeds into B(22 ⊕ 1).

Fig. 4. The closure algebra 4.

Lemma 8.10. Let V be a variety of closure algebras containing 4. Then 42 is a V-finitely presented unifiable algebra. Proof. Fact 2.5 yields that 42 is V-finitely presented. The reader may also see the appendix for an exact presentation of 42 . Moreover, there is a homomorphism from 4 onto 2. Hence 42 is unifiable. 2

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Fig. 5. The closure algebra B(22 ⊕ 1).

In our considerations the algebra B(22 ⊕ 1), depicted in Fig. 5, which is the free Boolean extension of the Heyting algebra 22 ⊕ 1, plays a crucial role. Note that B(22 ⊕ 1) is isomorphic to the modal algebra which is dual to the ordered set Lev2 treated as a modal frame. Lemma 8.11. Let U be a variety of closure McKinsey algebras containing B(22 ⊕ 1), W be a variety of monadic algebras and V = U ∨ W be their varietal join. Then 42 does not embed into FV . Proof. In order to obtain a contradiction, assume that 42 embeds into F. Then, since 42 is finite, it embeds into FV (k) for some natural number k. Recall that, by Theorem 8.8, FV (k) ∼ = FU (k) × GW (k). The congruence lattice of 42 is isomorphic to a product of two three-element chains. Let ρ be the congruence from the middle of this square, i.e., ρ is generated by ((a, a), (1, 1)). Now, if R is a simple closure algebra in W and h : 42 → R is a homomorphism, then the image of h is also simple and ρ  ker(h). Thus for every homomorphism h : 42 → GW (k) we also have ρ  ker(h). Moreover, ρ ∩ α = id42 iff α = id42 for every congruence α of 42 . These facts yield that 42 embeds into FU (k). Since 22  42 , 22 also embeds into FU (k). The rest of the proof is very similar to the proof of Lemma 7.14. The generator t of an isomorphic image of 22 in FV (k) satisfies 0 < t, ¬t < 1

and 2t = t,

2¬t = ¬t.

Therefore, by Lemma 8.4, there exists a homomorphism f : FU → 22 such that f (t) = (1, 0) and ¬t) = (0, 1). Let a = {(1, 0)}, b = {(0, 1)} be the open atoms and a∨ f (¬ ∨b = {(0, 1), (1, 0)} be the open coatom in B(22 ⊕ 1). Let g : V → B(22 ⊕ 1) be a mapping given by ⎧ ⎪ a if f (v) = (1, 0) ⎪ ⎪ ⎪ ⎨b if f (v) = (0, 1) g(v) = , ⎪ ∨b if f (v) = (1, 1) ⎪a∨ ⎪ ⎪ ⎩ 0 if f (v) = (0, 0) and g¯ : F → B(22 ⊕ 1) be the homomorphic extension of g. Let h : B(22 ⊕ 1) → 22 be the surjective homomorphism such that h(a) = (1, 0) and h(b) = (0, 1). Then f |V = h ◦ g. Thus, by the universal mapping ¬t) ∈ h−1 ((0, 1)) = {b, b∨ property of FV (k), f = h ◦ g¯. We get that g¯(t) ∈ h−1 ((1, 0)) = {a, a∨ ∨c} and g¯(¬ ∨c}, 2 where c is the third, closed atom of B(2 ⊕ 1). Therefore, since g¯ maps open elements onto open element, ¬t) = b. Now we compute in B(22 ⊕ 1) g¯(t) = a and g¯(¬ ¬t) = g¯(t)∨ ¬g¯(t) = a∨ 1 = g¯(1) = g¯(t∨ ∨¬ ∨¬ ∨b < 1, and reach a contradiction. We finish this proof with one technical remark. In the earlier version of this paper, instead of U we dealt with σ(Y), where Y was a variety of Heyting algebras containing 22 ⊕ 1. One would then wish to use Lemma 7.14 instead of repeating the whole argumentation as we did here. But it does not give a correct

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proof. It follows from the fact that in general FY (k) is not isomorphic to O(Fσ(Y) (k)). Actually, FY (k) only embeds into O(Fσ(Y) (k)) and this embedding is proper even in the simple case when Y = VLev2 . 2 Theorem 8.12. Let U be an SC variety of closure algebras containing B(22 ⊕ 1), for instance any of σ(VLev2 ), σ(VLev 3 ), . . . , σ(VMed ), W be a non-minimal nontrivial variety of monadic algebras and V = U ∨W be their varietal join. Then 1. V is ASC \SC, 2. V does not have unitary exact (and hence unitary neither exact nor projective) unification. Proof. (1) This is a consequence of Theorem 8.8. Notice that all varieties among σ(VLev2 ), σ(VLev 3 ), . . ., σ(VMed ) are SC. Indeed, it follows from Proposition 7.15 and the fact that σ operator preserves SC [78, Theorem 5.4.7]. (2) Since 4 embeds into B(22 ⊕ 1), 42 ∈ V. So the claim follows from Lemmas 8.10 and 8.11 and Proposition 5.2. The lack of unitary unification was also proved in [23, Lemma 4], where it was shown that every variety of closure algebras, not necessarily ASC, containing B(22 ⊕ 1) cannot have unitary unification. Again, then 22 is a V-finitely presented algebra without a most general unifier. 2 Let us note that the mapping (U, V) → U ∨ V, where U and V are nontrivial varieties satisfying the condition from Theorem 8.8, is injective. To see this assume that V = U 0 ∨ W 0 = U 1 ∨ W 1 , where U 0 , U 1 and W 0 , W 1 satisfy the same conditions as U and W respectively. By [50, Corollary 4.2], 0 0 1 1 VSI = USI ∪ WSI = USI ∪ WSI .

Since U i are varieties of McKinsey algebras and W j are varieties of monadic algebras, for i, j ∈ {0, 1}, we have U i ∩ V j = {2}. 0 1 0 1 This yields the equations USI = USI and VSI = VSI . Hence U 0 = U 1 and V 0 = V 1 . Therefore, there are at least as many ASC\SC varieties of closure algebras as there are SC varieties of closure algebras. However, we do not know exactly how many of them exist. We know that there are at least ℵ0 (the number of Levin varieties) and at most c (the number of varieties of closure algebras) members in each family. Note that there are continuum many ASC varieties of modal algebras [24, Corollary 14].

Problem 8.13. How many SC and ASC varieties of closure algebras exist? Acknowledgement We wish to thank Alex Citkin who suggested that the ASC\SC property might be detected among ASC varieties by identifying some simple algebra. We confirmed it for varieties of closure algebras in Proposition 8.7. We also would like to thank the reviewer for many valuable remarks, in particular, for drawing our attention to exact unification. Appendix A Fact A.1. (Cf. Lemma 8.10.) Let V be a variety of closure algebras containing 4. Then the algebra 42 is isomorphic to the V-finitely presented algebra P(232x≈32x)∧(32x∧ ∧x≈2x)∧(32x∨ ∨x≈3x) .

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Proof. Let α be the congruence of a free algebra FV (1) over {v} generated by the pairs e0 =(232v, 32v), ∧v, 2v), e1 =(32v∧ ∨v, 3v). e2 =(32v∨ The fact that e0 ∈ α guarantees that 32v/α is not just closed, but also an open element in FV (1)/α. Thus FV (1)/α is isomorphic to a product M0 × M1 such that 32a0 = 1 in M0 , 32a1 = 0 in M1 , where a0 and a1 are generators of M0 and M1 obtained by projecting v/α. Now e1 ∈ α yields that in M0 2a0 = 32a0 ∧ ∧a0 = a0 , 3a0 = 32a0 = 1 and hence 2¬a0 = 0, ¬a0 = ¬a0 . 3¬ In particular, M0 = {0, 1, a0 , ¬a0 } and M0 is a homomorphic image of 4. Now we compute in M1 . Since e1 ∈ α 2a1 = 32a1 ∧ ∧a1 = 0, and since e2 ∈ α 3a1 = 32a1 ∨ ∨a1 = a1 . Hence 2¬a1 = a1 , ¬a1 = 0 3¬ and M1 is also a homomorphic image of 4. Thus FV (1)/α is a homomorphic image of 42 . Now let h : FU (1) → 42 be a surjective homomorphism such that h(v) = (a, ¬a), where a is the open and ¬a is the closed atom in 4. A routine verification reveals that e0 , e1 , e2 ∈ ker(h). Thus FV (1)/α is isomorphic to 42 . 2 References [1] H. Andréka, B. Jónsson, I. Németi, Free algebras in discriminator varieties, Algebra Universalis 28 (3) (1991) 401–447. [2] H. Bass, Finite monadic algebras, Proc. Amer. Math. Soc. 9 (1958) 258–268. [3] C. Bergman, Structural completeness in algebra and logic, in: Algebraic Logic, Budapest, 1988, in: Colloq. Math. Soc. János Bolyai, vol. 54, North-Holland, Amsterdam, 1991, pp. 59–73. [4] W.J. Blok, Varieties of interior algebras, Ph.D. thesis, University of Amsterdam, 1976, http://www.illc.uva.nl/Research/ Dissertations/HDS-01-Wim_Blok.text.pdf. [5] W.J. Blok, P. Dwinger, Equational classes of closure algebras. I, Nederl. Akad. Wetensch. Proc. Ser. A, Indag. Math. 78 (3) (1975) 189–198. [6] W.J. Blok, D. Pigozzi, On the structure of varieties with equationally definable principal congruences. I, Algebra Universalis 15 (2) (1982) 195–227, http://dx.doi.org/10.1007/BF02483723. [7] W.J. Blok, D. Pigozzi, Algebraizable logics, Mem. Amer. Math. Soc. 77 (396) (1989), vi+78.

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