Algebraic expansions of logics and algebras and a case study of Abelian l-groups and perfect MV-algebras
Miguel Campercholi, Diego Castaño, José Patricio Díaz Varela, Joan Gispert
aa r X i v : . [ m a t h . L O ] J un Algebraic expansions of logics and algebras and acase study of Abelian ℓ -groups and perfectMV-algebras M. Campercholi, D. Casta˜no, J. P. D´ıaz Varela, J. GispertJune 18, 2020
Abstract
An algebraically expandable (AE) class is a class of algebraic struc-tures axiomatizable by sentences of the form @D ! ŹŹŹ p “ q . For a logic L algebraized by a quasivariety Q we show that the AE-subclasses of Q correspond to certain natural expansions of L , which we call algebraicexpansions . These turn out to be a special case of the expansions byimplicit connectives studied by X. Caicedo. We proceed to characterizeall the AE-subclasses of Abelian ℓ -groups and perfect MV-algebras, thusfully describing the algebraic expansions of their associated logics. The idea of expanding structures in a given language with new operations andrelations definable in some way is pervasive in Algebra and Model Theory. If wefocus on operations defined by systems of equations on algebraic structures wearrive at the notion of
Algebraic Expansions ([16]). Restricting to this kind ofdefinability has the advantage of producing well-behaved expansions that can bestudied with ‘universal-algebraic’ techniques (e.g., sheaf representations). Wedescribe these expansions in more detail.Let τ be an algebraic language. Given a class of τ -algebras K and a systemof equations of the form s p x , . . . , x n , z , . . . , z m q “ t p x , . . . , x n , z , . . . , z m q ... s k p x , . . . , x n , z , . . . , z m q “ t k p x , . . . , x n , z , . . . , z m q we can consider the class A of those algebras in K for which, given values for the x ’s, there are unique values for the z ’s such that all equalities hold. We say that A is an Algebraically Expandable (AE) subclass of K given that the membersof A can be expanded with the operations defined by the system of equations.1or example, let K be the class of tÑ , u -subreducts of Boolean algebras, andconsider the system of equations z Ñ x “ ,z Ñ x “ , pp x Ñ z q Ñ p x Ñ z qq Ñ p x Ñ z q “ . The class A in this case is the class of algebras in K where every two elementshave a meet with respect to the ordering induced by Ñ . The expansion of A is(term-equivalent to) the class of generalized Boolean algebras.In the setting of Abstract Logic expansions by new connectives are a com-mon theme as well, in particular, expansions of a logic L with connectives de-termined in some way by L . As we know, there is a long-standing and fruitfulinterplay between Logic and Algebra, so it is natural to consider what, if any,is the logical counterpart of AE-classes. As we shall see, for the case of analgebraizable logic L with equivalent algebraic semantics Q , the AE-subclasses Q are in correspondence with the family of a specific kind of expansions of L ,which we call algebraic expansions . The notion of an algebraic expansion of alogic turns out to be quite natural, we think, and interestingly it falls into thegeneral framework of expansions by implicit connectives studied by X. Caicedoin [9]. An immediate consequence is that algebraic expansions are again alge-braizable. The algebraic expansions of L are naturally ordered by morphismsthat preserve the language of L . It turns out that this is a lattice ordering whenconsidered modulo equipollency, and the ensuing lattice is dually isomorphicwith the lattice of AE-subclasses of Q under inclusion.Besides introducing the notion of algebraic expansions of a logic we analyzetwo particular cases: ℓ -groups and perfect MV-algebras. In both cases we obtainfull descriptions of the AE-classes, and thus, of the algebraic expansions oftheir corresponding logics. We show that in both cases there is a continuumof expansions, and the lattices are isomorphic with ω ‘ and ω ‘ , in theformer and latter case respectively.In the next section we summarize all the basic definitions and propertiesof the theory of AE-classes needed for this article. In Section 3 we give theformal definition of algebraic expansion of a logic, and prove the fundamentalresults linking them with AE-classes (Theorems 3.1 and 3.2). In Section 4 wecharacterize the AE-classes of Abelian ℓ -groups and the algebraic expansions oftheir corresponding logic. Finally, in Section 5, we translate the results fromSection 4 to their analogs for perfect MV-algebras, using cancellative hoops asan intermediate step. This completely describes the algebraic expansions of theassociated logic. In this section we introduce fundamental definitions, establish notation andpresent several basic facts needed in the sequel. We assume familiarity with2asic Universal Algebra, Model Theory and Abstract Algebraic Logic (see, e.g.,[8, 24, 20], respectively).
Throughout this article algebras are considered as models of first-order lan-guages without relations. For example, Abelian ℓ -groups are algebras in the lan-guage τ G : “ t` , ´ , , _ , ^u . As is customary we use bold letters ( A , B , C , . . . )for algebraic structures and italic letters ( A, B, C, . . . ) for the underlying sets.For algebras A and B we write A Ď B whenever A is a subalgebra of B .Given a structure A in a language τ and a term t p x , . . . , x n q in the samelanguage, we write t A p ¯ a q for the value of the term upon assigning elements a , . . . , a n from A to the variables x , . . . , x n . We may omit the superscript A if there is no risk of confusion.Given a (first-order) formula ϕ , we say that ϕ is • an identity if it has the form @ ¯ x p p p ¯ x q “ q p ¯ x qq , where p and q are terms, • a quasi-identity if it has the form @ ¯ x p α p ¯ x q ÑÑÑ β p ¯ x qq , where α is a finiteconjunction of term-equalities and β is a term-equality, • universal if it has the form @ ¯ xψ , where ψ is quantifier-free, • existential if it has the form D ¯ xψ , where ψ is quantifier-free.A sentence is a formula with no free variables. If Σ is a set of sentences, Mod p Σ q denotes the class of all models that satisfy the sentences in Σ.Whenever we consider a class K of algebras, we assume that all algebras in K have the same language. Given a class K of algebras, we define the usualclass operators: • I p K q denotes the class of isomorphic images of members of K , • H p K q denotes the class of homomorphic images of members of K , • S p K q denotes the class of subalgebras of members of K , • P p K q denotes the class of direct products with factors in K , • P U p K q denotes the class of ultraproducts with factors in K .If O is one of the above operators and K “ t A , . . . , A n u , we write O p A , . . . , A n q instead of O p K q .Remember that a class K is a variety (or equational class ) if it can be axiom-atized using a set of identities; equivalently, K is a variety if it is closed under H , S and P . The smallest variety containing a given class K is HSP p K q and isdenoted by V p K q . A quasivariety is a class of algebras axiomatized by a set ofquasi-identities. A class K is a quasivariety if and only if K is closed under I , S , P and P U ; the smallest quasivariety containing a given class K is ISPP U p K q ,also denoted by Q p K q . Finally, recall that K is universal if it is axiomatized byuniversal sentences, which is equivalent to K being closed under I , S and P U .Moreover, the smallest universal class containing a class K is given by ISP U p K q . We write the first-order connectives ^^^ , ___ , ÑÑÑ , ØØØ in bold font to distinguish them fromalgebraic operations and connectives in sentential logics. K and two sentences ϕ , ψ , we say that ϕ and ψ are equivalent in K , and write ϕ „ ψ in K , if for every A P K we have that A ( ϕ if and onlyif A ( ψ . In order to define algebraically expandable classes [16], one of the fundamentalnotions in this article, we need to define the special type of sentences that definethem. An equational function definition sentence (EFD-sentence for short) inthe language τ is a sentence of the form @ x . . . x n D ! z . . . z m k ľľľ i “ s i p ¯ x, ¯ z q “ t i p ¯ x, ¯ z q (1)where s i , t i are τ -terms, n ě
0, and m ě
1. Suppose ϕ is the EFD-sentencein (1). Observe that ϕ is valid in a structure A if and only if the system ofequations k ľľľ i “ s i p ¯ x, ¯ z q “ t i p ¯ x, ¯ z q defines a (total) function F : A n Ñ A m . Wedenote the coordinate functions of F by r ϕ s A , . . . , r ϕ s A m .Let ϕ be as in (1). We define: • E p ϕ q : “ @ ¯ x D ¯ z k ľľľ i “ s i p ¯ x, ¯ z q “ t i p ¯ x, ¯ z q , • U p ϕ q : “ @ ¯ x ¯ y ¯ z k ľľľ i “ s i p ¯ x, ¯ y q “ t i p ¯ x, ¯ y q ^^^ k ľľľ i “ s i p ¯ x, ¯ z q “ t i p ¯ x, ¯ z q ÑÑÑ ¯ y “ ¯ z .The following basic facts are used without explicit reference throughout thearticle. • ϕ is equivalent to E p ϕ q ^^^ U p ϕ q . • U p ϕ q is (equivalent to) a conjunction of quasi-identities. • E p ϕ q is preserved by homomorphic images, that is, for any surjectivehomomorphism f : A Ñ B , if A ( E p ϕ q , then B ( E p ϕ q .A class of algebras K is an algebraically expandable class (AE-class for short)if there is a set of EFD-sentences Σ such that K “ Mod p Σ q . Let K and C beclasses of algebras, K Ď C . We say that K is an AE-subclass of C if K isaxiomatizable by EFD-sentences relative to C , that is, K “ C X Mod p Σ q forsome set Σ of EFD-sentences. The reader should be aware that K may be anAE-subclass of C , but fail to be an AE-class itself.Let Q be a quasivariety in the language τ and let Σ be a set of EFD-sentences. There is an obvious expansion of the AE-subclass K : “ Q X Mod p Σ q of Q obtained by skolemizing the existential quantifiers in Σ; details follow. For4ach ϕ P Σ of the form @ x . . . x n D ! z . . . z m ŹŹŹ ki “ s i p ¯ x, ¯ z q “ t i p ¯ x, ¯ z q considernew n -ary function symbols f ϕ , . . . , f ϕm and the set of identities E ϕ : “ t@ ¯ x s i p ¯ x, f ϕ p ¯ x q , . . . , f ϕm p ¯ x qq “ t i p ¯ x, f ϕ p ¯ x q , . . . , f ϕm p ¯ x qq : 1 ď i ď k u . Let τ Σ be the expansion of τ obtained by adding the f ϕj ’s for each ϕ P Σ, andput E Σ : “ ď ϕ P Σ E ϕ ,U Σ : “ t U p ϕ q : ϕ P Σ u . Define Q Σ : “ Mod p Γ Y E Σ Y U Σ q , where Γ is a set of quasi-identities axiomatizing Q . We call Q Σ an algebraicexpansion of Q . Note that Q Σ is a quasivariety over the language τ Σ whosemembers are precisely the expansions of the members of K . An interestingproperty of this process is that, up to term-equivalence, the quasivariety Q Σ isdetermined by the AE-subclass K and not by the axiomatization Σ; for detailssee Theorem 3.2.We conclude this section with a preservation result for EFD-sentences neededin the sequel. Recall that a structure A is finitely subdirectly irreducible if itsdiagonal congruence is finitely meet-irreducible in the congruence lattice of A .We write K fsi for the class of finitely subdirectly irreducible members of K .A variety is arithmetical provided that it is both congruence distributive andcongruence permutable. Lemma 2.1.
Let V be an arithmetical variety such that V fsi Yt trivial algebras u isa universal class, and let A P V . If ϕ is an EFD-sentence such that H p A q fsi ( ϕ ,then A ( ϕ .Proof. By [23] A has a global representation with factors in H p A q fsi , and by[30] global representations preserve EFD-sentences. Throughout this work, by a (sentential) logic over a language τ we mean afinitary structural consequence operator on the set of τ -formulas. We referthe reader to [20] for definitions and results about abstract algebraic logic notexplicitly mentioned in this article.Let L be an algebraizable logic and let ∆ p x, y q be a set of equivalence formu-las for L . Given a finite set of L -formulas Φ p ¯ x, ¯ z q on variables x , . . . , x n , z , . . . , z m , n, m P ω , let f Φ1 , . . . , f Φ m be new n -ary symbols and let L Φ be the least proposi-tional logic containing L such that: $ L Φ Φ p ¯ x, f Φ1 p ¯ x q , . . . , f Φ m p ¯ x qq , (E Φ )Φ p ¯ x, ¯ y q , Φ p ¯ x, ¯ z q $ L Φ ∆ p ¯ y, ¯ z q . (U Φ )5∆ p ¯ y, ¯ z q is shorthand for Ť mj “ ∆ p y j , z j q .) We say that L Φ is the algebraic ex-pansion of L by Φ. Recall that if ∆ p x, y q is another set of equivalence formulasfor L , then ∆ p x, y q %$ L ∆ p x, y q . Thus the expansion L Φ does not depend onthe choice of equivalence formulas.Given a set Σ of finite sets of L -formulas, define L Σ as the least propositionallogic containing L Φ for every Φ P Σ. (Of course, we assume that the new symbolsfor each L Φ are different.) The logic L Σ is called the algebraic expansion of L by Σ.Observe that, in the definition of L Φ , for the case m “ Φ holdsvacuously, so L Φ is just the axiomatic extension of L by Φ. Hence, axiomaticextensions of L are algebraic expansions of L .As mentioned in the introduction, in [9] Caicedo studies expansions of finitelyalgebraizable logics where the behaviour of the new connectives is determinedby the added axioms and rules. We call such an expansion of a logic L an expansion of L by implicit connectives .It is easy to see that the expansion L Σ defined above is in fact an expansionof L by implicit connectives (where p E Φ q and p U Φ q correspond to new axiomsand rules, respectively). As an immediate consequence of this fact we have that L Σ is algebraizable with the same equivalence formulas and defining equationsas L [9, Theorem 1]. Furthermore, the equivalent algebraic semantics of L Σ isthe expected one [9, Corollary 2], which in this case turns out to be an algebraicexpansion of the equivalent algebraic semantics of L . Details follow.Let Q be the equivalent algebraic semantics of L via the set of equivalenceformulas ∆ p x, y q and the set of defining equations ε p x q . Given a finite set Φ p ¯ x, ¯ z q of L -formulas, let e p Φ q be the EFD-sentence @ ¯ x D !¯ z ŹŹŹ ε p Φ p ¯ x, ¯ z qq . For Σ a set offinite sets of L -formulas define e p Σ q : “ t e p Φ q : Φ P Σ u . Now, Corollary 2 in [9]says that the algebraic expansion Q e p Σ q is the equivalent algebraic semantics of L Σ . Thus, for each algebraic expansion of L we have a corresponding algebraicexpansion of Q . Of course, we can also go in the other direction. Given an EFD-sentence ϕ : “ @ ¯ x D !¯ z α p ¯ x, ¯ z q , put d p ϕ q : “ ∆ p α p ¯ x, ¯ z qq . Here and in the sequel∆ p α p ¯ x, ¯ z qq abbreviates Ť ki “ ∆ p s i p ¯ x, ¯ z q , t i p ¯ x, ¯ z qq if α p ¯ x, ¯ z q is the conjunction ofequations ŹŹŹ ki “ s i p ¯ x, ¯ z q “ t i p ¯ x, ¯ z q . For a set Σ of EFD-sentences, we write d p Σ q for the set t d p ϕ q : ϕ P Σ u . Again, it is straightforward to check that Q Σ is theequivalent algebraic semantics of L d p Σ q . Furthermore, • L Σ “ L d p e p Σ qq , • Q Σ “ Q e p d p Σ qq for suitable Σ’s. This establishes a direct correspondence between algebraicexpansions of a logic and those of its equivalent algebraic semantics. Theorem3.2 below explores this connection in greater detail. In the sequel, to avoidcumbersome notation, given a logic L and a set Σ of EFD-sentences we write L Σ instead of L d p Σ q .For future reference, the facts above are summarized in the following: To avoid confusion with the connectives of the logic we use Ź for the logical conjunction. heorem 3.1. Let L be a finitely algebraizable logic with equivalent algebraicsemantics Q . Let Σ be a set of EFD-sentences in the language of Q . Then thealgebraic expansion L Σ is algebraizable with the same equivalence formulas anddefining equations as L , and its equivalent algebraic semantics is the quasivari-ety Q Σ . Moreover, there is a one-to-one correspondence between the algebraicexpansions of L and the algebraic expansions of Q . We conclude this section with an example of a logic that has an expansionby implicit connectives that is not algebraic. Let L int be the Intuitionistic Logicand let L S int be the extension of L int by the implicit connective S defined in[11, Example 5.2]. The equivalent algebraic semantics of L S int is the variety H S of Heyting algebras with successor. It is not hard to show that the classof Heyting-reducts of algebras in H S is not an AE-subclass of H . Thus, byTheorem 3.1, L S int cannot be an algebraic expansion of L int . Let L be a finitely algebraizable logic with equivalent algebraic semantics Q . TheAE-subclasses of the quasivariety Q are naturally (lattice-)ordered by inclusion.In the current section we show how this ordering translates to the algebraicexpansions of Q , and thus to the algebraic expansions of L . For this we need tolook into interpretations between logics and between classes of algebras.Fix a countably infinite set of variables X : “ t x , x , . . . u ; given a language τ we write T m p τ q for the set of τ -terms over the variables in X . Let τ and τ be two expansions of a language τ . A τ -translation from τ into τ is a function T : τ Ñ T m p τ q such that T maps each symbol of arity n to a term in thevariables x , . . . , x n , and T p f q “ f p x , . . . , x n q for every n -ary symbol f P τ .Let K and K be two classes of algebras over τ and τ , respectively. A τ -interpretation of K in K is a τ -translation T : τ Ñ T m p τ q such that for everymember A : “ p A, t g A : g P τ uq in K , the algebra A T : “ p A, t T p f q A : f P τ uq belongs to K . If T and S are τ -interpretations of K in K and K in K ,respectively, such that the maps A ÞÑ A T and A ÞÑ A S are mutually inverse,we say that K and K are τ -term-equivalent .We turn now to maps between logics. A τ -translation T from τ into τ extends in a natural way to a mapping from T m p τ q to T m p τ q : • T p x q “ x for every variable x P X ; • T p f p ϕ , . . . , ϕ n qq “ T p f qp T p ϕ q , . . . , T p ϕ n qq for f in τ of arity n and ϕ , . . . ϕ n in T m p τ q .Given a set Γ of τ -terms we write T p Γ q for t T p ϕ q : ϕ P Γ u .Let τ and τ be expansions of a language τ , and suppose L and L arelogics in τ and τ , respectively. A τ -morphism from L to L is a τ -translationfrom τ into τ such thatΓ $ L ϕ implies T p Γ q $ L T p ϕ q Y t ϕ u Ď T m p τ q . We say that L and L are τ -equipollent (cf. [12])provided there are τ -morphisms T from L to L and S from L to L suchthat: • ϕ %$ L S p T p ϕ qq for every ϕ P T m p τ q ; • ϕ %$ L T p S p ϕ qq for every ϕ P T m p τ q .The following result shows the connection between the above defined rela-tions. Theorem 3.2.
Let L be a finitely algebraizable logic in the language τ withequivalent algebraic semantics Q . Let Σ and Σ be two sets of EFD-sentencesin τ .1. The following are equivalent: p i q there is a τ -morphism from L Σ to L Σ , p ii q there is a τ -interpretation of Q Σ in Q Σ , p iii q Q X Mod p Σ q Ď Q X Mod p Σ q .2. The following are equivalent: p i q L Σ and L Σ are τ -equipollent, p ii q Q Σ and Q Σ are τ -term-equivalent, p iii q Q X Mod p Σ q “ Q X Mod p Σ q .Proof. Fix a finite set of equivalence formulas ∆ p x, y q and a finite set of definingequations ε p x q that witness the algebraization relation between L and Q . FromTheorem 3.1 we know that the same sets algebraize L Σ and L Σ with Q Σ and Q Σ as their corresponding equivalent algebraic semantics.Let us start by proving 1. p i qñp ii q . Assume first there is a τ -morphism T from L Σ to L Σ . We provethat T is also a τ -interpretation of Q Σ in Q Σ . Let A P Q Σ ; we aim toprove that A T P Q Σ . Since the τ -reducts of A T and A coincide, the alge-bra A T satisfies the quasi-identities valid in Q . Let ϕ : “ @ ¯ x D !¯ z α p ¯ x, ¯ z q bean EFD-sentence in Σ . We show that A T satisfies the identities E ϕ and thequasi-identity U p ϕ q . We start by showing that A T satisfies E ϕ . By defini-tion, we have $ L Σ ∆ p α p ¯ x, f ϕ p ¯ x q , . . . , f ϕm p ¯ x qq and, since T is a τ -morphism, $ L Σ T p ∆ p α p ¯ x, f ϕ p ¯ x q , . . . , f ϕm p ¯ x qqq . Both ∆ and α are in the language τ , so $ L Σ ∆ p α p ¯ x, T p f ϕ qp ¯ x q , . . . , T p f ϕm qp ¯ x qqq . Then Q Σ ( @ ¯ x α p ¯ x, T p f ϕm qp ¯ x q , . . . , T p f ϕm qp ¯ x qq ,and thus A T ( @ ¯ x α p ¯ x, f ϕ p ¯ x q , . . . , f ϕm p ¯ x qq . This shows that A T ( E ϕ . We shownext that A T satisfies U p ϕ q . Again, by definition, we know that ∆ p α p ¯ x, ¯ y qq Y ∆ p α p ¯ x, ¯ z qq $ L Σ ∆ p ¯ y, ¯ z q . Applying the τ -morphism T to this deduction immedi-ately produces ∆ p α p ¯ x, ¯ y qq Y ∆ p α p ¯ x, ¯ z qq $ L Σ ∆ p ¯ y, ¯ z q since all formulas involvedare in T m p τ q . By the algebraization relation Q Σ ( U p ϕ q and, in particular, A ( U p ϕ q . Again, noting that A and A T have the same τ -reduct, we get that A T ( U p ϕ q . 8 ii qñp i q . Let T be a τ -interpretation from Q Σ in Q Σ . We claim that T is a τ -morphism from L Σ to L Σ . By the algebraizability relation, this amounts toshowing that ε p Γ q ( Q Σ ε p ϕ q implies ε p T p Γ qq ( Q Σ ε p T p ϕ qq ( ˚ )for every Γ Yt ϕ u Ď T m p τ Σ q . Since the defining equations ε p x q only use symbolsfrom τ , we have that p˚q is equivalent to ε p Γ q ( Q Σ ε p ϕ q implies T p ε p Γ qq ( Q Σ T p ε p ϕ qq . ( ˚˚ )Now, observe that for all algebras A P Q Σ , all τ Σ -terms t p ¯ x q and all tuples ¯ a from A we have that T p t q A p ¯ a q “ t A T p ¯ a q . From this fact it is straightforward toprove p˚˚q . p ii qñp iii q . Suppose T is a τ -interpretation from Q Σ in Q Σ , and let A P Q X Mod p Σ q . Let A Σ the expansion of A in Q Σ . Then p A Σ q T P Q Σ . Since p A Σ q T satisfies Σ and Σ is a set of τ -sentences, A satisfies Σ as well. Thisshows that A P Q X Mod p Σ q . p iii qñp ii q . This follows from the proof of [13, Theorem 5].We turn now to the equivalences in 2. p i qñp iii q . This is immediate from 1. p iii qñp ii q . Assume Q X Mod p Σ q “ Q X Mod p Σ q . From 1. there are τ -interpretations T and S from Q Σ in Q Σ and from Q Σ in Q Σ , respectively. Fix A in Q Σ , and let f be a symbol in τ Σ z τ . By the definition of Q Σ , there isan EFD-sentence ϕ : “ @ ¯ x D !¯ z α p ¯ x, ¯ z q in Σ such that f “ f ϕi . Put B : “ p A T q S and note that it suffices to prove that f A “ f B . Fix a sequence ¯ a of elementsfrom A . Since B ( E ϕ , we have that B ( α p ¯ a, p f ϕ q B p ¯ a q , . . . , p f ϕm q B p ¯ a qq . As A and B have the same τ -reduct and α is a τ -formula, it follows that A ( α p ¯ a, p f ϕ q B p ¯ a q , . . . , p f ϕm q B p ¯ a qq . We also know that A ( E ϕ , and thus A ( α p ¯ a, p f ϕ q A p ¯ a q , . . . , p f ϕm q A p ¯ a qq . Since A ( U p ϕ q , we conclude that p f ϕi q A p ¯ a q “p f ϕi q B p ¯ a q . We proved that p A T q S “ A . Analogously p A S q T “ A for every A P Q Σ . p ii qñp i q . Suppose Q Σ and Q Σ are τ -term-equivalent and let T and S be τ -interpretations such that p A S q T “ A for every A P Q Σ and p A T q S “ A forevery A P Q Σ . We claim that T and S make L Σ and L Σ τ -equipollent. By1., the maps T and S are τ -morphisms from L Σ in L Σ and from L Σ to L Σ ,respectively. It remains to show that ϕ %$ L Σ S p T p ϕ qq for every ϕ P T m p τ Σ q and ϕ %$ L Σ T p S p ϕ qq for every ϕ P T m p τ Σ q . We prove the first equivalence,the second one being analogous. By the algebraizability relation, it is enoughto prove that ε p ϕ q )( Q Σ ε p S p T p ϕ qqq , or equivalently, ε p ϕ q )( Q Σ S p T p ε p ϕ qqq .In fact, we claim that γ )( Q Σ S p T p γ qq for every τ Σ -equation γ . Indeed, forany A in Q Σ and any tuple ¯ a from A we have that A ( γ p ¯ a q iff p A S q T ( γ p ¯ a q iff A S ( T p γ qp ¯ a q iff A ( S p T p γ qqp ¯ a q .Let L be a logic algebraized by a quasivariety Q . As is the case for anyquasivariety, the AE subclasses of Q form a lattice Λ under inclusion. In thelight of Theorem 3.2, the algebraic expansions of L modulo equipollency and9rdered by morphisms form a lattice as well, dually isomorphic with Λ. Thus,classifying the algebraically expandable classes of Q yields a classification of allalgebraic expansions of L up to equipollency. When the AE-subclasses of a quasivariety are known, Theorem 3.2 immediatelygives the description of the algebraic expansions of the corresponding logic. Wepresent here three examples.
An algebra A is called primal if it is finite and every function f : A n Ñ A for n ě A . It is proved in [16] that the only AE-subclassesof V p A q for a primal A are V p A q and the class of trivial algebras. Thus, the only(modulo equipollency) algebraic expansions of a logic L algebraized by such avariety are L itself and the inconsistent logic. This applies, e.g., to ClassicalPropositional Logic and m -valued Post’s logic. Recall that Gdel Logic L G is the extension of Intuitionistic Logic by the pre-linearity axiom p x Ñ y q _ p y Ñ x q . It is known that the equivalent algebraicsemantics of L G is the variety H G of Gdel algebras, also known as prelinearHeyting algebras. The only AE-subclasses of H G are its subvarieties ([14]).Thus, the algebraic expansions of L G agree with its axiomatic extensions. Let L Ñ be the implicative fragment of classical propositional logic. The equiv-alent algebraic semantics of L Ñ is the variety I of implication algebras. Recallthat disjunction is expressible in terms of Ñ , thus for n ě ď i ď ns ni p x , . . . , x n q : “ n ł j “ ,j ‰ i x j is an tÑu -term. For each n ě n : “ t z Ñ s ni p ¯ x q : i P t , . . . , n uu Y t n ł i “ p s ni p ¯ x q Ñ z qu . By definition, L Φ n Ñ is the least expansion of L Ñ that satisfies p E Φ n q and p U Φ n q .However, condition p U Φ n q is already true for L Ñ . Thus L Φ n Ñ is the expansion of L Ñ by the following axioms µ n p ¯ x q Ñ s ni p ¯ x q for i P t , . . . , n u , Ž ni “ p s ni p ¯ x q Ñ µ n p ¯ x qq , 10here µ n is a new n -ary symbol.By the characterization of the AE-subclasses of I given in [15] it followsfrom Theorem 3.2 that, up to equipollency, the consistent algebraic expansionsof L Ñ are L Ñ ă . . . ă L Φ Ñ ă L Φ Ñ where L ă L means that there is an tÑu -morphism from L to L but L and L are not equipollent. Observe that µ is the classical conjunction and, moregenerally, we have that µ n p ¯ x q “ Ź ni “ s ni p ¯ x q .Example 3 of [9] shows classical negation is implicitly definable in L Ñ . Sincenone of the algebraic expansions of L Ñ has classical negation as a term, we haveanother example of an expansion by implicit connectives that is not algebraic. ℓ -groups andthe Logic of Equilibrium In this section we give a complete description of the AE-classes of Abelian ℓ -groups. In particular, we show that they form a lattice isomorphic with ‘ ω (where ‘ denotes the ordinal sum). In view of Theorem 3.2 this produces acomplete characterization of the algebraic expansions of the Logic of Equilibrium([21, 26]).Recall that an Abelian ℓ -group is a structure in the language τ G : “ t` , ´ , , _ , ^u such that: • p A, ` , ´ , q is an Abelian group, • p A, _ , ^q is a lattice, • a ` p b _ c q “ p a ` b q _ p a ` c q for every a, b, c P A .Clearly Abelian ℓ -groups form a variety, which we denote by G . We write G to todenote its subclass of totally ordered members. Since all ℓ -groups in this articleare Abelian, we sometimes omit the word Abelian.In the following lemma we collect some properties that are needed in thesequel. Lemma 4.1.
1. The variety G is arithmetical, that is, every member of G has permutableand distributive congruences.2. For every nontrivial A P G to we have ISP U p A q “ G to .3. An Abelian ℓ -group is finitely subdirectly irreducible if and only if it isnontrivial and totally ordered.4. For every nontrivial A P G we have Q p A q “ V p A q “ G .Proof. Since ℓ -groups have both group and lattice reducts, it is clear that theyhave permutable and distributive congruences (see [8, Section II.12]). From11he proof of Theorem 4 in [31], it follows that every finitely generated totallyordered Abelian ℓ -group is embeddable in an ultrapower of Z , the ℓ -group ofintegers. This implies that G to “ ISP U p Z q , so 2. follows from the fact that Z is a subalgebra of any nontrivial ℓ -group. Item 3. is proved in [22, Lemma3.5.4]. We prove 4.; fix a nontrivial A in G and note that Q p A q Ď V p A q Ď G .Since Z is a substructure of A , we have that G to “ ISP U p Z q Ď Q p A q . So ISP p G to q Ď ISP p Q p A qq “ Q p A q and, as 3. says that G “ ISP p G to q , we aredone. ℓ -groups We proceed to characterize EFD-sentences modulo equivalence in G . We first re-duce the problem to totally ordered Abelian ℓ -groups and then show the specialrole that divisible groups play as regards EFD-sentences. Lemma 4.2.
Given EFD-sentences ϕ, ψ , if ϕ „ ψ in G to , then ϕ „ ψ in G .Proof. Suppose ϕ „ ψ in G to ; take a nontrivial A in G , and assume A ( ϕ .On the one hand, since U p ϕ q is a quasi-identity, Lemma 4.1.(4) implies that H p A q ( U p ϕ q . On the other hand, H p A q ( E p ϕ q because E p ϕ q is preservedby homomorphic images. Hence H p A q ( ϕ and, in particular, H p A q fsi ( ϕ .As, by Lemma 4.1.(3), every member in H p A q fsi is totally ordered, we have H p A q fsi ( ψ . So, using Lemma 2.1, we are done.For each positive integer k define δ k : “ @ x D ! z kz “ x. Our next step is to show that every EFD-sentence is equivalent to a δ k in G ,which is accomplished in Theorem 4.12.Recall that an ℓ -group G is divisible if for every g P G and every positiveinteger n , there exists h P G such that g “ nh . Given a divisible ℓ -group D ,since ℓ -groups are torsion-free, we have that δ k holds D for all k ; thus, we candefine the expansion D : “ p D , pr δ k s D q k ě q . The next result shows that the only functions defined by EFD-sentences inthese expansions are term-operations.
Theorem 4.3.
Let D be a totally ordered divisible ℓ -group and let ϕ be anEFD-sentence that holds in D . Then, the functions r ϕ s D , . . . , r ϕ s D m defined by ϕ on D are term-functions on D . The above theorem can be derived from [10, Theorem 20]. We provide adifferent proof that relies on the characterization of existentially closed algebrasin G to .Given a class K of algebras closed under isomorphisms and A P K , we saythat A is existentially closed in K if for every B P K such that A Ď B , everyexistential formula ϕ p ¯ x q , and every ¯ a P A n B ( ϕ p ¯ a q implies A ( ϕ p ¯ a q . ℓ -groups. Proposition 4.4.
Given a totally ordered ℓ -group G , we have that G is exis-tentially closed in G to if and only if G is divisible.Proof. The result follows from [29, Theorem 3.1.13] when considering totallyordered ℓ -groups as structures on a purely relational language τ . However,since the operations ` , ´ , _ , ^ are definable by quantifier-free τ -formulas, thestatement follows. Corollary 4.5.
Let D Ď G be totally ordered ℓ -groups and assume D is divis-ible. Then, for every EFD-sentence ϕ we have that G ( ϕ implies D ( ϕ .Proof. Suppose G satisfies the EFD-sentence ϕ . Since U p ϕ q is universal, wehave D ( ϕ , and the fact that D is existentially closed implies D ( E p ϕ q . Corollary 4.6. If ϕ is an EFD-sentence with a nontrivial model in G , thenevery totally ordered divisible ℓ -group satisfies ϕ .Proof. Assume H is a nontrivial model of ϕ and let H be a nontrivial totallyordered homomorphic image of H . Clearly H ( E p ϕ q and, since Q p H q is theclass of all ℓ -groups, we have H ( U p ϕ q . Hence H ( ϕ . By Lemma 4.1, weknow that ISP U p H q “ G to . Thus, if D is a totally ordered divisible ℓ -group,there is G P P U p H q such that D Ď G . Finally, Corollary 4.5 yields D ( ϕ .After this sequence of results we are ready to present: Proof of Theorem 4.3.
Assume D ( ϕ for some EFD-sentence ϕ , D nontrivial.Observe that V p D q is arithmetical since arithmeticity is witnessed by a Pixleyterm (see [8]).We prove first that V p D q fsi ( ϕ . Since all divisions are basic operations of D , we have that the algebras in SP U p D q are totally ordered divisible ℓ -groups,and Corollary 4.6 produces SP U p D q ( ϕ . Clearly HSP U p D q ( E p ϕ q and, since G “ Q p D q ( U p ϕ q , it follows that HSP U p D q satisfies U p ϕ q as well. Thus, HSP U p D q ( ϕ , and we are done since V p D q fsi Ď HSP U p D q by Jnsson’s lemma(see [25]).Since ℓ -group congruences are compatible with division operations, we havethat the congruences of algebras in V p D q agree with the congruences of their ℓ -group reducts, and so, V p D q fsi is a universal class. Thus, by Lemma 2.1, V p D q ( ϕ , and the conclusion follows now from [13, Lemma 3].Given a positive integer k and a term t p ¯ x q in τ G , let δ k,t : “ @ ¯ x D ! z kz “ t p ¯ x q . Observe that U p δ k,t q is valid in G because Abelian ℓ -groups are torsion-free.We denote by D the class of expansions D of divisible ℓ -groups D P G . Wewrite τ D for the language of the algebras in the class D .13 emma 4.7. Given a term s p ¯ x q in τ D , there is a term t p ¯ x q in τ G and a positiveinteger k such that k s p ¯ x q “ t p ¯ x q is valid in D . Hence, for any divisible D P G the term-function s D agrees with the function r δ k,t s D .Proof. It follows by induction on the structure of s p ¯ x q . Lemma 4.8.
Let ϕ be an EFD-sentence with a nontrivial model in G . Thenthere are positive integers k , . . . , k m and terms t , . . . , t m in τ G such that ϕ „ m ľľľ j “ δ k j ,t j in G .Proof. Fix ϕ : “ @ x . . . x n D ! z . . . z m α p ¯ x, ¯ z q . Note that G ( U p ϕ q since ϕ hasa nontrivial model in G and G has no proper subquasivarieties. Let D be anontrivial totally ordered divisible ℓ -group. By Corollary 4.6, we have that D ( ϕ . So Theorem 4.3 provides terms s , . . . , s m in τ D such that r ϕ s D j “ s D j for j P t , . . . , m u . Moreover, by Lemma 4.7, there are positive integers k , . . . , k m and terms t , . . . , t m in τ G such that s D j “ r δ k j ,t j s D . This shows that D ( @ ¯ x ¯ z p α p ¯ x, ¯ z q ØØØ m ľľľ j “ k j z j “ t j p ¯ x qq , and again using that G has no propersubquasivarieties, we have G ( @ ¯ x ¯ z p α p ¯ x, ¯ z q ØØØ m ľľľ j “ k j z j “ t j p ¯ x qq . Finally, since G satisfies U p ϕ q and U p δ k j ,t j q for j P t , . . . , m u , it follows that ϕ „ m ľľľ j “ δ k j ,t j in G . In the following, by a system of linear inequalities we mean a finite con-junction of inequalities of the form a x ` ¨ ¨ ¨ ` a n x n ě a , . . . , a n areintegers. (Note that such a system can be written as a conjunction of equationsin τ G .)We say that a system of linear inequalities α p ¯ x q is full-dimensional on anAbelian ℓ -group G if there is no p a , . . . , a n q P Z n zt ¯0 u such that G ( @ ¯ x p α p ¯ x qÑÑÑ ř a i x i “ q . That is, the system α p ¯ x q imposes no linear dependencies on its so-lutions in G . Observe that Lemma 4.1.(4) implies that α p ¯ x q is full-dimensionalon some nontrivial ℓ -group G if and only if it is full-dimensional on every non-trivial ℓ -group. Hence, we say that α p ¯ x q is full-dimensional provided it is full-dimensional on some nontrivial ℓ -group. Lemma 4.9.
A system of inequalities α p ¯ x q is full-dimensional if and only iffor every totally ordered ℓ -group G the set t ¯ g P G n : G ( α p ¯ g qu generates G n as an Abelian group.Proof. Assume α p ¯ x q is a full-dimensional system of inequalities and let S G : “t ¯ g P G n : G ( α p ¯ g qu for any totally ordered ℓ -group G . Let Q and Z denotethe ℓ -groups of rational and integer numbers, respectively. First observe that14 Z “ S Q X Z n . Note also that S G is closed under linear combinations whose co-efficients are non-negative integers, and S Q is closed under non-negative rationallinear combinations.We start by proving that S Z generated Z n as Abelian group. Let V be the Q -vector subspace of Q n generated by S Q . Observe that V “ Q n ; otherwise, therewould exist integers a , . . . , a n , not all zero, such that V Ď t ¯ x P Q n : ř a i x i “ u , contradicting the fact that α p ¯ x q is full-dimensional. Since V “ Q n , thesolution set S Q contains a Q -basis of Q n , which, multiplied by a suitable positiveinteger, yields a Q -basis t ¯ b , . . . , ¯ b n u Ď S Z . Since S Z is closed under positiveinteger linear combinations, ¯ b : “ ř ¯ b i P S Z . Now, let ¯ c P Z n be arbitrary andwrite ¯ c “ ř r i ¯ b i for suitable rational numbers r i . Let k be a positive integersuch that k ě ´ r i for all i . Then k ¯ b ` ¯ c “ ř i p k ` r i q ¯ b i P S Q , since it is apositive linear combination of elements in S Q . Thus k ¯ b ` ¯ c P S Q X Z n “ S Z ,and so ¯ c “ p k ¯ b ` ¯ c q ´ k ¯ b belongs to the Abelian group generated by S Z .We prove now that S G generates G n as an Abelian group for any totallyordered group G . For any ¯ a P Z n and g P G we write ¯ ag : “ p a g, . . . , a n g q .Note that if ¯ a P S Z and g is a non-negative member of G , then ¯ ag P S G . Fix j P t , . . . , n u , and let ¯ e j P Z n be such that e ji “ i “ j and e ji “ e j “ ř k l ¯ a l for integers k l and ¯ a l P S Z . Hence, if g P G , g ě
0, then ¯ e j g “ ř k l ¯ a l g is an integer linear combination of solutions ¯ a l g P S G .This proves that S G generates ¯ e j g for all j and all g P G , g ě
0. Now it followseasily that any ¯ g P G n is generated by elements in S G .The converse implication is straightforward. Lemma 4.10.
Let t p ¯ x q be a term in τ G . There are full-dimensional systemsof linear inequalities α p ¯ x q , . . . , α m p ¯ x q and terms t p ¯ x q , . . . , t m p ¯ x q , which areinteger linear combinations of the variables x , . . . , x n , such that for all G P G to and all ¯ g P G n we have t G p ¯ g q “ $’’&’’% t G p ¯ g q if α p ¯ g q , ... t G m p ¯ g q if α m p ¯ g q . (2) Proof.
Fix a τ G -term t p ¯ x q . We show first that there are full-dimensional systems α p ¯ x q , . . . , α m p ¯ x q and Abelian group terms t p ¯ x q , . . . , t m p ¯ x q such that (2) holdsfor G “ R , the ℓ -group of real numbers.Using the way the lattice and group operations interact, we may assume t p ¯ x q “ s p u p ¯ x q , . . . , u p p ¯ x qq where u p ¯ x q , . . . , u p p ¯ x q are Abelian group terms (i.e.,linear combinations of variables with integer coefficients) and s p ¯ y q is a latticeterm. For each permutation σ of t , . . . , p u let α σ p ¯ x q be the system of linear in-equalities expressing that u σ p q p ¯ x q ď ¨ ¨ ¨ ď u σ p p q p ¯ x q . Since R is totally ordered,for each σ there is j σ P t , . . . , p u such that t R p ¯ r q “ u R j σ p ¯ r q for all ¯ r such that α σ p ¯ r q . Next, for each σ let S σ : “ t ¯ r P R n : α σ p ¯ r qu . As each ¯ r P R n satisfies atleast one α σ p ¯ x q , we have that R n “ Ť σ S σ . Let t σ , . . . , σ m u be the set of15ermutations σ such that S σ has nonempty interior. Note that α σ j p ¯ x q is full-dimensional on R for all j P t , . . . , m u (and thus on every ℓ -groups). Sinceeach S σ is a closed subset of R n , by a simple topological argument, we have R n “ S σ Y ¨ ¨ ¨ Y S σ m . So, defining α j p ¯ x q : “ α σ j p ¯ x q and t j p ¯ x q : “ u σ j p ¯ x q for j P t , . . . , m u , we haveestablished (2) in the case G “ R . To conclude we show that the same α j ’s and t j ’s work for any G P G to . In fact, note that (2) holds if and only if G satisfiesthe following universal formulas • @ ¯ x p α j p ¯ x q ÑÑÑ t p ¯ x q “ t j p ¯ x qq for j P t , . . . , m u , • @ ¯ x p α p ¯ x q ___ . . . ___ α m p ¯ x qq .Since these formulas hold in R , Lemma 4.1.(2) says that they must hold in G . Lemma 4.11.
Given a positive integer k and an τ G -term t , there is a positiveinteger k such that δ k,t „ δ k in G .Proof. Fix a positive integer k and an τ G -term t ; let α j p ¯ x q and t j p ¯ x q for j Pt , . . . , m u be as in Lemma 4.10. Suppose t j p ¯ x q “ a j x `¨ ¨ ¨` a jn x n , and let d bethe greatest common divisor of the set t k uY t a ji : i P t , . . . , n u , j P t , . . . , m uu .Define k by k “ dk ; we prove that δ k,t „ δ k in G . Observe that, due to Lemma4.2, it suffices to show that δ k,t „ δ k in G to .Take G P G to and assume G ( δ k,t . We claim that t j p ¯ g q is divisible by k forevery ¯ g P G n and j P t , . . . , m u . Indeed, given ¯ g P G n and j P t , . . . , m u , byLemma 4.9, we can write ¯ g “ ř b l ¯ g l for some integers b l and some ¯ g l P G n suchthat G ( α j p ¯ g l q . Note that t p ¯ g l q “ t j p ¯ g l q for each l . Since G ( δ k,t , for each l there is h l P G such that kh l “ t p ¯ g l q “ t j p ¯ g l q . Thus t j p ¯ g q “ t j p ÿ b l ¯ g l q“ ÿ b l t j p ¯ g l q“ ÿ b l kh l “ k ÿ b l h l , which proves the claim.Now write d “ kc ` ř i,j a ji c ji for suitable integers c and c ji . Then, for any g P G , dg “ kcg ` ÿ j ÿ i a ji c ji g “ kcg ` ÿ j t j p ¯ g j q where ¯ g j : “ p c j g, . . . , c jn g q . Since each t j p ¯ g j q is divisible by k , it follows thatthere is g P G such that dg “ kg . Thus dg “ dk g , so d p g ´ k g q “ G is torsion-free, g “ k g . This proves that G ( δ k .16onversely, assume any element in G is divisible by k , and fix ¯ g : “ p g , . . . , g n q P G n . We prove that t p ¯ g q is divisible by k . Let j P t , . . . , m u be such that t p ¯ g q “ t j p ¯ g q . Since each a ji is divisible by d , there is g P G such that t j p ¯ g q “ dg .Now, since g is divisible by k , there is g P G such that g “ k g . Putting alltogether we obtain t p ¯ g q “ t j p ¯ g q “ dg “ dk g “ kg .We are now ready to present our characterization of EFD-sentences in G . Theorem 4.12.
Given an EFD-sentence ϕ with a nontrivial model in G thereis a positive integer k such that ϕ „ δ k in G .Proof. Given ϕ , combining Lemmas 4.8 and 4.11, we have that there are positiveintegers k , . . . , k m such that ϕ „ m ľľľ j “ δ k j in G . Now take k : “ k ¨ ¨ ¨ k m , andnote that m ľľľ j “ δ k j is equivalent to δ k in G .Given a set S of prime numbers, let Σ S : “ t δ p : p P S u . Since for an ℓ -groupdivisibility by k is equivalent to divisibility by the prime factors of k , we havethe following: Theorem 4.13.
Every set of EFD-sentences either has only trivial models oris equivalent over G to Σ S for some set S of prime numbers. Furthermore,the map S ÞÑ Σ S is one-to-one, and thus, the lattice of AE-subclasses of G isisomorphic with ‘ ω . As shown in [21] the variety G of Abelian ℓ -groups is the equivalent algebraicsemantics of the Logic of Equilibrium Bal defined by the following:Axioms p ϕ Ñ ψ q Ñ pp θ Ñ ϕ q Ñ p θ Ñ ψ qqp ϕ Ñ p ψ Ñ θ qq Ñ p ψ Ñ p ϕ Ñ θ qqpp ϕ Ñ ψ q Ñ ψ q Ñ ϕ pp ψ Ñ ϕ q ` Ñ p ϕ Ñ ψ q ` q Ñ p ϕ Ñ ψ q ϕ `` Ñ ϕ ` Rules ϕ, ϕ Ñ ψ $ ψϕ, ψ $ ϕ Ñ ψϕ $ ϕ ` p ϕ Ñ ψ q ` $ p ϕ ` Ñ ψ ` q ` The derived connectives:0 : “ x Ñ x , ´ x : “ x Ñ x ` y : “ ´ x Ñ y , x _ y : “ p x Ñ y q ` ` x , x ^ y : “ ´p´ x _ ´ y q . 17orm a complete set for Bal since x Ñ y %$ ´ x ` y and x ` %$ x _
0. This allowsus to say that G is the equivalent algebraic semantics of Bal via equivalenceformulas ∆ p x, y q “ t x Ñ y u and defining equations ε p x q “ t x “ u .Given a prime number p , the algebraic expansion of Bal corresponding tothe EFD-sentence δ p is, by definition, obtained from Bal by adding the axiom: x Ñ p d p p x q ( A p )and the rule U t x Ñ pz u . Since this rule is derivable in Bal , the expansion isobtained simply by adding A p . For a set S of prime numbers define Bal S asthe expansion of Bal by the axioms t A p : p P S u . Note that, since Bal S isan axiomatic expansion of Bal , its equivalent algebraic semantics is a variety.These expansions were also considered in [10] where it is proved that everyimplicit connective in the logic
Bal
P rimes is explicit.Recall that an expansion L of a logic L is called conservative provided thatfor each set of L -formulas Γ Y t ϕ u we have that Γ $ L ϕ implies Γ $ L ϕ . Theorem 4.14.
1. Every algebraic expansion of
Bal is τ G -equipollent to exactly one of thefollowing: • Inconsistent Logic, • Bal S for some set S of prime numbers.2. The algebraic expansions of Bal form a lattice isomorphic with ω ‘ when ordered by τ G -morphisms.3. Given sets S, S of prime numbers with S Ď S , the expansion Bal S isconservative over Bal S .Proof. Items 1. and 2. follow from Theorems 3.2 and 4.13. We prove 3.Fix sets of prime numbers S Ď S , and let V and V be the equivalentalgebraic semantics of Bal S and Bal S , respectively. Since Bal S is finitary, toprove 3. it is enough to show that any quasi-identity in the language of V validin V is also valid in V . Let Q S be the ℓ -group of rational numbers expandedwith the divisions by the primes in S . It is not hard to show that Q S generates V as a quasivariety, that is, Q p Q S q “ V . Now let ϕ be a quasi-identity in thelanguage of V that is valid in V . Then, we have that Q S ( ϕ , and thus, Q S ( ϕ . Since Q p Q S q “ V , the proof is finished. In this section we characterize the AE-subclasses of the variety V p P q , where P is the class of perfect MV-algebras. So we also obtain a full description of thealgebraic expansions of L P , the Logic of Perfect MV-Algebras (see, e.g., [3]).18ur approach is to export the results for Abelian ℓ -groups to perfect MV-algebras, exploiting the connection between these two classes (see [28, 27, 18]).We first translate the classification of EFD-sentences given in Theorem 4.12 tocancellative hoops —the positive cones of Abelian ℓ -groups. These structuresprovide a stepping stone to carry our results over to perfect MV-algebras. Given an Abelian ℓ -group G , its positive cone is the subset G ` : “ t x P G : x ě u . We define the algebraic structure G ` : “ p G ` , ` , ´ , q where x ´ y : “p x ´ y q_
0. The lattice structure of G ` (as a sublattice of G ) is determined by theoperations ` and ´ . Indeed, for every x, y P G ` we have that x _ y “ x `p y ´ x q and x ^ y “ x ´ p x ´ y q .We write H for the class of positive cones of Abelian ℓ -groups consideredas algebras in the language τ H : “ t` , ´ , u . The members of H are known as cancellative hoops ; see e.g. [19, 5]. Given a cancellative hoop A , there is (up toisomorphism) a unique Abelian ℓ -group whose positive cone is isomorphic with A (see [4]); we write A for this ℓ -group. Moreover, A is totally ordered if andonly if A is. We write H to for the class of totally ordered cancellative hoops. Lemma 5.1.
1. The class H is an arithmetical variety.2. For every nontrivial A P H , we have Q p A q “ V p A q “ H .3. A cancellative hoop is finitely subdirectly irreducible if and only if it isnontrivial and totally ordered.Proof. These results follow from the general theory of hoops. See [6, 7, 5,19].
Lemma 5.2.
Given EFD-sentences ϕ, ψ , if ϕ „ ψ in H to , then ϕ „ ψ in H .Proof. The proof is analogous to the one for Lemma 4.2 using Lemmas 2.1 and5.1.Our next step is to provide a translation of EFD-sentences in τ H into EFD-sentences in τ G . First, for an τ H -term t p ¯ x, ¯ z q define recursively the τ G -term t ˚ p ¯ x, ¯ z q by • ˚ : “ • x ˚ i : “ x i _ ´ x i • z ˚ j : “ z j • p t ` t q ˚ : “ t ˚ ` t ˚ • p t ´ t q ˚ : “ p t ˚ ´ t ˚ q _
0. 19ext, given a conjunction of equations α p ¯ x, ¯ z q : “ k ľľľ i “ t i p ¯ x, ¯ z q “ s i p ¯ x, ¯ z q in thelanguage τ H , define the τ G -formula α ˚ p ¯ x, ¯ z q : “ k ľľľ i “ t ˚ i p ¯ x, ¯ z q “ s ˚ i p ¯ x, ¯ z q . Finally, if ϕ : “ @ ¯ x D !¯ z α p ¯ x, ¯ z q is an EFD-sentence in the language τ H , we definethe translation of ϕ into τ G as the sentence ϕ ˚ : “ @ ¯ x D !¯ z α ˚ p ¯ x, ¯ z q ^^^ z ě ^^^ . . . ^^^ z m ě . The next lemma shows that our translations work as intended.
Lemma 5.3.
Let A P H to , let α p ¯ x, ¯ z q be a conjunction of τ H -equations, and let ϕ be an EFD-sentence in the language τ H .1. For all ¯ a, ¯ b from A we have A ( α p ¯ a, ¯ b q if and only if A ˚ ( α ˚ p ¯ a, ¯ b q .2. A ( ϕ if and only if A ˚ ( ϕ ˚ .Proof. To prove 1. it suffices to show that t A p ¯ a, ¯ b q “ t ˚ A ˚ p ¯ a, ¯ b q for all ¯ a, ¯ b from A , which is a easy induction on the structure of t p ¯ x, ¯ z q .Next, we prove the left-to-right implication of 2. Assume ϕ : “ @ ¯ x D !¯ z α p ¯ x, ¯ z q is valid in A . Given c , . . . , c n P A ˚ , for each i let a i : “ c i _ ´ c i (note that a i P A ). There are b , . . . , b m P A such that A ( α p ¯ a, ¯ b q ; thus, by item 1., A ˚ ( α ˚ p ¯ a, ¯ b q . By the definition of α ˚ , this is equivalent to A ˚ ( α ˚ p ¯ c, ¯ b q ;hence A ˚ ( E p ϕ ˚ q . To prove the uniqueness part suppose there are c , . . . , c n , b , . . . , b m , b , . . . , b m P A ˚ such that A ˚ ( α ˚ p ¯ c, ¯ b q ^^^ α ˚ p ¯ c, ¯ b q and b i ě b i ě i . If for each i we take a i : “ c i _ ´ c i , then we have A ˚ ( α ˚ p ¯ a, ¯ b q ^^^ α ˚ p ¯ a, ¯ b q , so A ( α p ¯ a, ¯ b q ^^^ α p ¯ a, ¯ b q . Thus ¯ b “ ¯ b , and we have shown A ˚ ( ϕ ˚ .The right-to-left implication is straightforward and left to the reader.From Section 4 recall that δ k is the sentence @ x D ! z kz “ x . Note that δ k isan EFD-sentence in both τ H and τ G . Lemma 5.4.
For all positive integers k we have δ ˚ k „ δ k in G .Proof. By Lemma 4.2, it suffices to check the equivalence of δ k and δ ˚ k in G to .Let G P G to be such that G ( δ ˚ k , i.e., G satisfies @ x D ! z p kz “ x _ ´ x ^^^ z ě q . Given a P G , there is b ě G such that kb “ a _ ´ a . If a ě
0, then kb “ a . Otherwise, a ď kb “ ´ a , so k p´ b q “ a . This shows that G ( δ k .Conversely, assume G ( δ k . Given a P G , there is b P G such that kb “ a _ ´ a .Since kb ě
0, it follows that b ě
0. Thus G ( δ ˚ k .20e are now ready to prove a characterization of EFD-sentences for H anal-ogous to Theorem 4.12. Theorem 5.5.
Given an EFD-sentence ϕ with a nontrivial model in H thereis a positive integer k such that ϕ „ δ k in H .Proof. Assume ϕ has a nontrivial model in H . By Lemma 5.3.2., the sentence ϕ ˚ has a nontrivial model in G . Thus, by Theorem 4.12 and Lemma 5.4, thereis a positive integer k such that ϕ ˚ „ δ ˚ k in G . Now, for every A P H to we have A ( ϕ ô A ˚ ( ϕ ˚ ô A ˚ ( δ ˚ k ô A ( δ k . Thus ϕ „ δ k in H to and, as a consequence, also in H . The class of MV-algebras is the equivalent algebraic semantics of Lukasiewiczinfinite-valued logic and has been extensively studied [17]. We recall next itsdefinition and some basic facts.An
MV-algebra is a structure A in the language τ MV : “ t` , , u such that: • p A, ` , q is an Abelian monoid, • x “ x , • x ` “ • p x ` y q ` y “ p y ` x q ` x .We write MV for the class of MV-algebras. Given A P MV we define theoperations _ and ^ by x _ y : “ p x ` y q ` y and x ^ y : “ p x _ y q . As iswell-known, p A, _ , ^ , , q is a bounded distributive lattice whose underlyingpartial ordering is given by x ď y if and only if x ` y “
0. Another relevantderived operation on A is ˚ , which is defined by x ˚ y : “ p x ` y q . It isalso well-known that p A, ˚ , q is an Abelian monoid. We define multiples andpowers of a P A recursively by: • a : “ a and a : “ a . • p n ` q a : “ na ` a and a n ` : “ a n ˚ a for any positive integer n .Let A be an MV-algebra. An ideal of A is a nonempty down-set that isclosed under ` . The radical of A is the intersection of all maximal ideals of A ; we denote it by rad A . We say that A is perfect if it is nontrivial and A “ rad A Y rad A where rad A : “ t a : a P rad A u (this definition isequivalent to the original one given in [2], see Corollary 4.5 in that reference).The class of perfect MV-algebras is denoted by P . We shall need the fact thatrad A “ t a P A : a “ u for A P P (see [17, Prop. 3.6.4]). We denote thetwo-element MV-algebra by ; clearly P P . Let P to denote the class of totallyordered perfect MV-algebras.As in the previous sections, the following two lemmas provide an essentialtool for our classification of EFD-sentences.21 emma 5.6.
1. The variety V p P q is arithmetical.2. An algebra in V p P q is finitely subdirectly irreducible if and only if it is atotally ordered perfect MV-algebra.Proof. These results follow from the general theory of MV-algebras. See [17,18].
Lemma 5.7.
Given EFD-sentences ϕ, ψ , if ϕ „ ψ in P to , then ϕ „ ψ in V p P q .Proof. The proof is analogous to the one for Lemma 4.2 using Lemmas 2.1 and5.6.Next we define a set of EFD-sentences that behave in a special way withrespect to the radical. An EFD-sentence ϕ : “ @ ¯ x D !¯ z α p ¯ x, ¯ z q in the language τ MV belongs to Φ rad if and only if for every A P P to and every ¯ a P A n thefollowing holds: p i q if ¯ a P p rad A q n and A ( α p ¯ a, ¯ b q for some ¯ b P A m , then ¯ b P p rad A q m , p ii q if ¯ a R p rad A q n we have that A ( α p ¯ a, ¯0 q and ¯ z “ ¯0 is the unique solutionto α p ¯ a, ¯ z q in A . Lemma 5.8.
Given an EFD-sentence ϕ in τ MV with a model in P , there areEFD-sentences ϕ , . . . , ϕ r P Φ rad such that ϕ „ r ľľľ i “ ϕ i in P to .Proof. We introduce some useful notation. Given ¯ e : “ p e , . . . , e n q P t , u n and an n -tuple of variables ¯ x : “ p x , . . . , x n q , define ¯ x ¯ e : “ p w , . . . , w n q where w i : “ x i if e i “ w i : “ x i if e i “
1. We also use this notation for n -tuplesof elements from an MV-algebra A : if ¯ a P A n , then ¯ a ¯ e : “ ¯ b where b i : “ a i if e i “ b i : “ a i if e i “
1. Finally define ρ p ¯ x q : “ x “ ^^^ . . . ^^^ x n “ , r ρ p ¯ x q : “ p x ^ ¨ ¨ ¨ ^ x n q “ . Note that given A P P and ¯ a P A n we have A ( ρ p ¯ a q if and only if each a i P rad A , and r ρ p ¯ x q is equivalent over P to the negation of ρ p ¯ x q .Assume ϕ : “ @ ¯ x D !¯ z α p ¯ x, ¯ z q has a model in P . Since is both a subalgebraand a quotient of any member of P , we have that ( ϕ . Thus, given ¯ e P t , u n ,there is a unique ¯ e P t , u m such that α p ¯ e, ¯ e q holds. Let η ¯ e p ¯ x, ¯ z q be the formula p ρ p ¯ x q ^^^ α p ¯ x ¯ e , ¯ z ¯ e qq or p r ρ p ¯ x q ^^^ ¯ z “ ¯0 q . Now observe that on any totally ordered (perfect) MV-algebra the disjunction x “ y or z “ w
22s equivalent to the equation pp x ˚ y q ` p y ˚ x qq ^ pp z ˚ w q ` p w ˚ z qq “ . Thus there is a conjunction of equations α ¯ e p ¯ x, ¯ z q equivalent to η ¯ e p ¯ x, ¯ z q in P to .Finally define ϕ ¯ e : “ @ ¯ x D !¯ z α ¯ e p ¯ x, ¯ z q . We claim that ϕ is equivalent in P to to the conjunction of all ϕ ¯ e for ¯ e P t , u n ,and that each of these formulas belongs to Φ rad .Fix A P P to and assume A ( ϕ . Let ¯ e P t , u n and ¯ a P A n . We showthat there is ¯ b P A m such that A ( α ¯ e p ¯ a, ¯ b q . If A ( ρ p ¯ a q , take ¯ c as the unique m -tuple such that A ( α p ¯ a ¯ e , ¯ c q . Then A ( ρ p ¯ a q ^^^ α p ¯ a ¯ e , p ¯ c ¯ e q ¯ e q since p ¯ c ¯ e q ¯ e “ ¯ c ,so we can take ¯ b : “ ¯ c ¯ e . If A * ρ p ¯ a q , take ¯ b : “ ¯0. It is easy to see that ¯ b isunique and so we have that A ( ϕ ¯ e .Conversely, assume A ( ϕ ¯ e for every ¯ e P t , u n and fix ¯ a P A n . Thereis ¯ e P t , u n such that ¯ a ¯ e P p rad A q n , that is, such that A ( ρ p ¯ a ¯ e q . Since A ( ϕ ¯ e , there is ¯ b P A m such that A ( α pp ¯ a ¯ e q ¯ e , ¯ b ¯ e q , that is, A ( α p ¯ a, ¯ b ¯ e q .This shows that A ( E p ϕ q . To check that A ( U p ϕ q , suppose ¯ c P A m is suchthat A ( α p ¯ a, ¯ c q . Since A ( U p ϕ ¯ e q and A ( α pp ¯ a ¯ e q ¯ e , p ¯ c ¯ e q ¯ e q , we have ¯ b “ ¯ c ¯ e ,and thus ¯ c “ ¯ b ¯ e .It remains to show that every ϕ ¯ e P Φ rad . Let B P P to and take ¯ a P p rad B q n ,¯ b P B m such that B ( α ¯ e p ¯ a, ¯ b q . Note that B ( α p ¯ a ¯ e , ¯ b ¯ e q . If h : B Ñ is the homomorphism with kernel rad B , it follows that ( α p ¯ e, h p ¯ b q ¯ e q . So,as ( U p ϕ q , we obtain h p ¯ b q ¯ e “ ¯ e . Hence h p ¯ b q “ ¯0, that is, ¯ b P p rad B q n .Condition p ii q in the definition of Φ rad holds by construction of α ¯ e p ¯ x, ¯ z q .Next we define a family of EFD-sentences that serve the same purpose asthe δ k ’s in our previous section. For each positive integer k define the τ MV -term t k p z q : “ p kz ^ z q _ z k and the EFD-sentence ε k : “ @ x D ! z t k p z q “ x. In the following lemma we collect several properties of these terms and EFD-sentences.
Lemma 5.9.
Let A P P .1. For every a P A we have t A k p a q “ a k if a P rad A ,ka if a P rad A .
2. The term-function t A k is a one-to-one endomorphism of A .3. The following are equivalent: i q t A k is surjective, p ii q A ( ε k , p iii q For every a P rad A , there is b P rad A such that kb “ a . p iv q For every a P rad A , there is b P rad A such that b k “ a .4. If A ( ε k , then r ε k s A is an automorphism, which is the inverse of t A k .Proof. Item 1. follows directly from the fact that 2 a “ a P rad A and2 a “ a P rad A .To prove 2. we show first that t A k preserves ` , that is, t A k p a ` b q “ t A k p a q ` t A k p b q . We consider three different cases and use item 1. in each case. If a, b P rad A , condition t A k p a ` b q “ t A k p a q ` t A k p b q reduces to k p a ` b q “ ka ` kb , whichholds in any MV-algebra. If a, b P rad A , we prove that p a ` b q k “ a k ` b k .Indeed, this equation holds since x ` y “ x, y P rad A . Finally, if a P rad A and b P rad A , we show that p a ` b q k “ ka ` b k . Indeed, in anyMV-algebra k p b _ a q “ k p b q _ ka . Now k p b _ a q “ k p p b ` a q ` a q “ k p p a ` b q ` a q “ k p a ` b q ` ka and k p b q _ ka “ p k b ` ka q ` ka “ p ka ` b k q ` ka . This shows that k p a ` b q ` ka “ p ka ` b k q ` ka and,since p rad A , `q is a cancellative semigroup ([18, Lemma 3.2]), we concludethat k p a ` b q “ p ka ` b k q , so p a ` b q k “ ka ` b k as was to be proved. The factthat t A k preserves 0 and is straightforward. Thus, t A k is an endomorphism of A . To show that it is one-to-one, it is enough to prove that t A k p a q “ a “
0. Indeed, if a P rad A , then t A k p a q “ ka “
0, so a ď ka “
0; if a P rad A ,then t A k p a q “ a k P rad A and cannot equal 0.Item 3. is a direct consequence of 1. and 2., and 4. follows easily from 2.and 3.Mundici’s functor Γ [17] allows us to make explicit the connection between δ k , defined in the previous section, and ε k . Given A P P , there is an Abelian ℓ -group G such that A – Γ p Z ~ ˆ G , p , qq where Z is the ℓ -groups of integersand ~ ˆ is the lexicographic product (note that G ` – rad A ). Now, Lemma5.9.3. p iii q says that A ( ε k if and only if G ( δ k .Given a model A of ε k we write d k for the function r ε k s A . In view of Lemma5.9.1. we see that d k is the analogue of division by k in ℓ -groups. To illustratethe behaviour of these functions we look at a concrete case. Example 5.10.
Let Q be the ℓ -group of rational numbers and consider theperfect MV-algebra D : “ Γ p Z ~ ˆ Q , p , qq . Recall that the universe of D is tp , x q : x P Q, x ě u Y tp , x q : x P Q, x ď u . It is straightforward tocheck that D ( ε k for every positive integer k and that t D k p i, x q “ p i, kx q and d D k p i, x q “ p i, xk q for every p i, x q from D .Next we show how the results for cancellative hoops translate to perfectMV-algebras. Given A P MV , we define x ´ y : “ p x ` y q . τ H -terms in MV-algebras. The radical of A is closed under ` and ´ . Moreover, rad A : “ p rad A , ` , ´ , q is a cancellativehoop (see [18, Lemma 3.2]). Note that with these definitions it is obvious thatfor an τ H -term t p ¯ x q and ¯ a P p rad A q n we have t A p ¯ a q “ t rad A p ¯ a q .Recall from Section 4.1 that δ k : “ @ x D ! z kz “ x . Lemma 5.11.
For every positive integer k and every A P P we have A ( ε k ô rad A ( δ k . Proof.
The equivalence of p ii q and p iii q in Lemma 5.9.3. shows that A ( E p ε k q ô rad A ( E p δ k q . Since U p ε k q is valid in P and U p δ k q is valid in H , the lemma now follows. Lemma 5.12.
For each ϕ P Φ rad with a model in P , there is an EFD-sentence ϕ H in the language τ H such that for every A P P to A ( ϕ ô rad A ( ϕ H . Proof.
Let ϕ : “ @ ¯ x D !¯ z α p ¯ x, ¯ z q in Φ rad . We can assume α p ¯ x, ¯ z q is a conjunctionof equations of the form t p ¯ x, ¯ z q “
0, where t is an τ MV -term. As in the proofs ofthe previous lemmas, since ϕ has a model in P , we know that ( ϕ . Moreover,since ϕ P Φ rad , we have ( t p ¯0 , ¯0 q “
0. Thus, by [1, Theorem 3.1], there isan τ H -term t p ¯ x, ¯ z q such that MV ( @ ¯ x ¯ z t p ¯ x, ¯ z q “ t p ¯ x, ¯ z q . Let β p ¯ x, ¯ z q be theresult of replacing each τ MV -term in α p ¯ x, ¯ z q by an equivalent τ H -term; define ϕ H : “ @ ¯ x D !¯ z β p ¯ x, ¯ z q .Fix A P P to . Suppose A ( ϕ and take ¯ a P p rad A q n . Since ϕ P Φ rad , thereis ¯ b P p rad A q m such that A ( α p ¯ a, ¯ b q , and thus rad A ( β p ¯ a, ¯ b q . Furthermore,if ¯ c P p rad A q m is such that rad A ( β p ¯ a, ¯ c q , then A ( α p ¯ a, ¯ c q , and it followsthat ¯ c “ ¯ b . This completes the proof of rad A ( ϕ H .For the other direction assume rad A ( ϕ H and let ¯ a P A n . If ¯ a R p rad A q n ,the definition of Φ rad implies that there is a unique ¯ b such that A ( α p ¯ a, ¯ b q ,namely ¯ b “ ¯0. To conclude, suppose ¯ a P p rad A q n . Since rad A ( ϕ H , thereis ¯ b P p rad A q m such that rad A ( β p ¯ a, ¯ b q , and thus A ( α p ¯ a, ¯ b q . If ¯ c P A m is such that A ( α p ¯ a, ¯ c q , then as ϕ P Φ rad we know that ¯ c P p rad A q m . Since rad A ( U p ϕ H q , it follows that ¯ c “ ¯ b .Recall that the identity @ x x “ x axiomatizes the class of Boolean algebrasrelative to the class of MV-algebras. Thus the only model of this identity in P is the two-element MV-algebra. Lemma 5.13.
Given ϕ P Φ rad with a model in P either ϕ „ @ x x “ x in P to or there is a positive integer k such that ϕ „ ε k in P to .Proof. If is the only model of ϕ in P to , then ϕ „ @ x x “ x in P to . Assume ϕ has a model A P P to non-isomorphic with . Then ϕ H has rad A as a nontrivialmodel (see Lemma 5.12). By Theorem 5.5, there is a positive integer k such25hat ϕ H „ δ k in H . From these facts and Lemma 5.11, for every B P P to wehave B ( ϕ ô rad B ( ϕ H ô rad B ( δ k ô B ( ε k . We are now in the position to prove a characterization of EFD-sentences forthe variety V p P q . Theorem 5.14.
For every EFD-sentence ϕ in τ MV with a model in P either ϕ „ @ x x “ x in V p P q or there is a positive integer k such that ϕ „ ε k in V p P q .Proof. By Lemma 5.8, there are basic EFD-sentences ϕ , . . . , ϕ r such that ϕ „ r ľľľ i “ ϕ i in P to .First suppose that ϕ has a model in P non-isomorphic with ; then so doeseach ϕ i . By Lemma 5.13, there are positive integers k , . . . , k r such that ϕ i „ ε k i in P to for every i P t , . . . , r u . Thus ϕ „ r ľľľ i “ ε k i in P to . Now take k : “ k . . . k r and note that r ľľľ i “ ε k i „ ε k in P to . Hence ϕ „ ε k in P to . Finally, by Lemma 5.7,we get that ϕ „ ε k in V p P q .Now, if is the only model of ϕ in P , then ϕ „ @ x x “ x in P to . Note that @ x x “ x is equivalent to the EFD-sentence @ x D ! z p x “ x q ^^^ p z “ x q , so wecan apply Lemma 5.7 to conclude that ϕ „ @ x x “ x in V p P q .As in the case of ℓ -groups, the characterization of EFD-sentences easily pro-vides a description of the AE-classes. Given a set S of prime numbers, letΣ S : “ t ε p : p P S u . Theorem 5.15.
Every set of EFD-sentences in τ MV is equivalent over V p P q to exactly one of the following: • t@ xy x “ y u , • t@ x x “ x u , • Σ S for some set S of prime numbers.Furthermore, the map S ÞÑ Σ S is one-to-one, and thus, the lattice of AE-subclasses of V p P q is isomorphic with ‘ ω . roof. From Lemma 5.9.3. it is easy to see that ε k is equivalent over V p P q to t ε p : p prime divisor of k u . This fact together with Theorem 5.14 proves thefirst part the the theorem.Now, given a set S of positive primes, if we consider the ℓ -group G of rationalnumbers whose denominators are products of primes in S , then for every prime p we have that the algebra Γ p Z ~ ˆ G , p , qq satisfies ε p if and only if p P S . Finally,observe that Boolean algebras trivially satisfy ε p for every prime number p . Thisproves the furthermore part. L P The Logic L P of Perfect MV-Algebras [18] is the extension of Lukasiewicz Logicby the axiom 2 x Ø p x q (recall that x Ø y : “ p x ` y q ^ p y ` x q ). As thename suggests, the equivalent algebraic semantics of L P is the variety V p P q .Given a prime number p , the algebraic expansion of L P corresponding tothe EFD-sentence ε p is, by definition, obtained from L P by adding the axiom: pp p d p p x q ^ d p p x q q _ d p p x q p q Ø x, ( D p )and the rule U tpp kz ^ z q_ z k qØ x u . Since this rule is derivable in L P , the expan-sion is obtained simply by adding D p .For a set S of prime numbers define L S P as the expansion of L P by the axioms t D p : p P S u . Note that, by the comment above, L S P is the algebraic expansionof L P corresponding to the AE-class axiomatized by Σ S : “ t ε p : p P S u . Thus,the equivalent algebraic semantics V p P q Σ S of L S P is a variety. Theorem 5.16.
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