Model completions for universal classes of algebras: necessary and sufficient conditions
aa r X i v : . [ m a t h . L O ] F e b T he J ournal of S ymbolic L ogic Volume 00, Number 0, XXX 0000
MODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS:NECESSARY AND SUFFICIENT CONDITIONS
GEORGE METCALFE AND LUCA REGGIO
Abstract.
Necessary and su ffi cient conditions are presented for the (first-order) theory of a universal class ofalgebraic structures (algebras) to admit a model completion, extending a characterization provided by Wheeler.For varieties of algebras that have equationally definable principal congruences and the compact intersectionproperty, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) byGhilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices,the existence of a model completion implies that the variety has equationally definable principal congruences.This result is then used to provide necessary and su ffi cient conditions for the existence of a model completion fortheories of Hamiltonian varieties of pointed residuated lattices, a broad family of varieties that includes lattice-ordered abelian groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of pointed residuatedlattices admits a model completion, it must have equationally definable principal congruences. In particular,the theories of lattice-ordered abelian groups and MV-algebras do not have a model completion, as first provedby Glass and Pierce, and Lacava, respectively. Finally, it is shown that certain varieties of pointed residuatedlattices generated by their linearly ordered members, including lattice-ordered abelian groups and MV-algebras,can be extended with a binary operation in order to obtain theories that do have a model completion. §1. Introduction. The main aim of this paper is to understand what it means inalgebraic terms for the (first-order) theory of a universal class of algebraic structures(algebras) to admit a model completion. For classes that have finite presentations —including all quasivarieties, but not, for example, ordered abelian groups — a completecharacterization was provided by Wheeler in [34] using the well-studied properties ofamalgamation and coherence together with a more complicated property referred to asthe conservative congruence extension property. However, as we show in Section 3,replacing coherence and the conservative congruence extension property by a variableprojection property and conservative model extension property, respectively, providesnecessary and su ffi cient conditions for all universal classes of algebras (Theorem 3.3).Although the mentioned properties can be used to confirm that the theories of orderedabelian groups and linearly ordered MV-algebras have a model completion [32, 26], theconservative model extension property is, in general, rather di ffi cult to prove or refute.We therefore also provide in Section 4 a more elegant characterization for varieties(equational classes) of algebras that have equationally definable principal congruencesand the compact intersection property, where the conservative model extension propertyis replaced by a more amenable equational variable restriction property (Theorem 4.5). Supported by Swiss National Science Foundation grant 200021 184693 and the European Union’s Horizon2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement 837724. © / / / $00.00 GEORGE METCALFE AND LUCA REGGIO
This result generalizes slightly a characterization given by Ghilardi and Zawadowskiin [16] (see also [17]) by covering also varieties such as lattice-ordered abelian groupsfor which there exists no equation that entails all other equations.In Section 5, we show that for congruence distributive varieties with the congruenceextension property satisfying a “guarded” deduction theorem, the existence of a modelcompletion implies that the variety has equationally definable principal congruences(Theorem 5.7). Our approach is inspired by the work of Glass and Pierce on lattice-ordered abelian groups in [19] and indeed yields both their result that the theory ofthis variety does not have a model completion, and the same result for MV-algebras,first proved by Lacava in [25]. More generally, we use Theorem 5.7 in Section 6 toshow that the theory of a Hamiltonian variety of pointed residuated lattices — a broadfamily of varieties that spans varieties of algebras for substructural logics as well aslattice-ordered abelian groups and MV-algebras (see, e.g., [5, 13, 28]) — has a modelcompletion if, and only if, the variety is coherent and has equationally definable prin-cipal congruences, the amalgamation property, and the equational variable restrictionproperty (Theorem 6.6).Finally, in Section 7, we associate with any variety V of pointed residuated latticesthat is generated by its linearly ordered members, a variety V ⊲ of algebras with anadditional binary operation that has equationally definable principal congruences andthe same universal theory as V in the original language. We then show that if V admitsa certain syntactic property, the theory of V ⊲ has a model completion (Theorem 7.5).Notably, this is the case for lattice-ordered abelian groups and MV-algebras, the secondcase yielding an alternative proof of the fact that the variety of MV ∆ -algebras has amodel completion, first announced by X. Caicedo at a conference in 2008. §2. Algebraic properties. Let us first recall some elementary material on universalalgebra, referring to [7] for proofs and references. For convenience, we will assumethroughout this paper that L is an algebraic language (i.e., a first-order language withno relation symbols) containing at least one constant symbol c and that an L -algebra A is an L -structure with universe A , where A is called trivial if | A | =
1. The set Con A ofcongruences of an L -algebra A always forms a complete lattice ordered by inclusion,and we let Cg A ( S ) denote the congruence of A generated by a set S ⊆ A .The term algebra Tm L ( x ) for L over a set x is an L -algebra, whose universe is the setTm L ( x ) of L -terms with variables in x . For any L -algebra A and map f : x → A , thereexists a unique homomorphism ˜ f : Tm L ( x ) → A extending f . Atomic L -formulas are L -equations , written s ≈ t , where s and t are L -terms. We call L -equations and theirnegations L -literals , and use the symbols &, g , ¬ , → , and ↔ to denote conjunctions,disjunctions, negations, implications, and bi-implications of L -formulas, respectively,defining ⊤ ≔ c ≈ c and ⊥ ≔ ¬⊤ . An L -formula π → ε , where π is a conjunction of L -equations and ε an L -equation, is an L -quasiequation . For an L -term t or L -formula α ,we denote by t ( x ) or α ( x ) that its free variables belong to the set x , and for a conjunctionof L -literals ξ , we write ξ + and ξ − for the conjunctions of L -equations occurring in ξ positively and negatively, respectively, assuming that the empty conjunction is ⊤ .Let H , I , S , P , and P U denote the class operators of taking homomorphic images,isomorphic images, subalgebras, products, and ultraproducts, respectively. A class of L -algebras K is called a variety if it is closed under H , S , and P , and a quasivariety if isclosed under I , S , P , and P U . The class K is a variety if, and only if, it is an equational ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS K is auniversal class if, and only if, it is closed under I , S , and P U , and a positive universalclass if, and only if, it is closed under H , S , and P U .The variety and quasivariety generated by a class of L -algebras K are, respectively,the smallest variety HSP ( K ) and quasivariety ISPP U ( K ) of L -algebras containing K . An L -equation is valid in K if, and only if, it is valid in HSP ( K ), and an L -quasiequation isvalid in K if, and only if, it is valid in ISPP U ( K ). For future reference, let us note alsothat any class of L -algebras K that is closed under taking finite products satisfies thefollowing disjunction property : for any conjunctions of L -equations ϕ, π , . . . , π n , K | = ϕ → g ≤ i ≤ n π i ⇐⇒ K | = ϕ → π i for some i ∈ { , . . . , n } .For a variety of L -algebras V , the V -free algebra F V ( x ) over a set x may be identifiedwith the quotient Tm L ( x ) /θ V ( x ), where θ V ( x ) ≔ T { θ ∈ Con Tm L ( x ) | Tm L ( x ) /θ ∈ V } .For any L -equation ε ∈ Tm L ( x ) , we let b ε denote its image under the natural surjection Tm L ( x ) ։ F V ( x ) . The following useful lemma shows that the valid L -quasiequationsof V can be described in terms of congruences of V -free algebras.L emma L -algebras V , conjunction of L -equations π ( x ), and L -equation ε ( x ), V | = π → ε ⇐⇒ b ε ∈ Cg F V ( x ) ( { b σ | σ is an equation of π } ) . In what follows, we will omit mention of the language L , assuming throughout thata class of algebras K is a class of L -algebras, and that terms, equations, formulas, etc.are defined over this language. Let us say that a class ofalgebras K has the variable projection property if for any finite set x , y and conjunctionof equations ϕ ( x , y ), there exists a quantifier-free formula ξ ( x ) such that K | = ϕ → ξ andfor any equation ε ( x ), K | = ϕ → ε = ⇒ K | = ξ → ε. If ξ ( x ) is required to be a conjunction of equations for each ϕ ( x , y ), we say that K hasthe equational variable projection property .R emark K satisfies the same quasiequations as the quasivariety ISPP U ( K )that it generates, K has the equational variable projection property if, and only if, ISPP U ( K ) has this property.For varieties, the variable projection property is equivalent both to the equationalvariable projection property and to the widely studied algebraic property of coherence.A variety V is said to be coherent if every finitely generated subalgebra of a finitelypresented member of V is finitely presented.P roposition V :(1) V is coherent.(2) V has the equational variable projection property.(3) V has the variable projection property.P roof . The equivalence of (1) and (2) follows directly from [24, Theorem 2.3], and(3) is an immediate consequence of (2). To show that (3) implies (2), we fix a finite set GEORGE METCALFE AND LUCA REGGIO x , y and a conjunction of equations ϕ ( x , y ), and let ξ ( x ) be a quantifier-free formula suchthat V | = ϕ → ξ and for any equation ε ( x ), we have V | = ϕ → ε = ⇒ V | = ξ → ε . As ϕ is satisfiable in V (e.g., in a trivial algebra), we can assume without loss of generalitythat ξ = ξ g · · · g ξ m , where ξ , . . . , ξ m are conjunctions of literals satisfiable in V .Since V | = ϕ → ξ , also V | = ϕ → g ≤ i ≤ m ξ + i and so, by the disjunction propertyfor varieties, there is an i ∈ { , . . . , m } such that V | = ϕ → ξ + i . Let σ , . . . , σ n be theequations of ξ − i and consider any equation ε ( x ) such that V | = ϕ → ε . By assumption, V | = ξ → ε , so V | = ξ i → ε . But then, by the disjunction property for varieties appliedto V | = ξ + i → ε g g ≤ j ≤ n σ j , either V | = ξ + i → ε or V | = ξ i → ⊥ . Since ξ i is satisfiablein V , we obtain V | = ξ + i → ε . ⊣ Following this last proposition, we will refer to a variety throughout this paper ascoherent whenever it has the (equational) variable projection property.R emark A is locally finite if every finitely generated subalgebra of A is finite, and a class of algebras K is locally finite if each A ∈ K is locally finite. Sinceany finitely presented algebra of a locally finite variety is finite and any finite algebra ofa variety is finitely presented, it follows that every locally finite variety is coherent.Below we introduce some well-known classes of algebras that will be employed asrunning examples throughout the paper. These algebras all possess definable binaryoperations ∧ and ∨ such that a ≤ b : ⇐⇒ a ∧ b = a defines a lattice order with binarymeets and joins given by ∧ and ∨ , respectively.E xample Linear orders with endpoints may be considered as bounded lattices h L , ∧ , ∨ , , i where the defined lattice order is linear. The class DL c of linear orderswith endpoints is then a positive universal class of algebras that generates the variety DL of bounded distributive lattices as a quasivariety. Since DL is locally finite, it is coherent,and, by Remark 2.2, the class DL c has the equational variable projection property.E xample Heyting algebra is an algebra h H , ∧ , ∨ , ⊃ , , i such that the reduct h H , ∧ , ∨ , , i is a bounded distributive lattice and ⊃ is the right residual of ∧ ; that is, a ≤ b ⊃ c if, and only if, a ∧ b ≤ c for all a , b , c ∈ H . Coherence of the variety HA of Heyting algebras is a consequence of Pitts’ uniform interpolation theorem forintuitionistic propositional logic [31].E xample lattice-ordered abelian group is an algebra h L , ∧ , ∨ , + , − , i such that h L , + , − , i is an abelian group, h L , ∧ , ∨i is a lattice, and a ≤ b implies a + c ≤ b + c for all a , b , c ∈ L . Lattice-ordered abelian groups form a variety LA that is coherent (see [24])and generated as a quasivariety by the positive universal class LA c of ordered abeliangroups , i.e., the class of linearly ordered members of LA (see, e.g., [1]). It follows fromRemark 2.2 that LA c has the equational variable projection property.E xample MV-algebra is an algebra h M , ⊕ , ¬ , i satisfying the equations(MV1) x ⊕ ( y ⊕ z ) ≈ ( x ⊕ y ) ⊕ z (MV4) ¬¬ x ≈ x (MV2) x ⊕ y ≈ y ⊕ x (MV5) x ⊕ ¬ ≈ ¬ x ⊕ ≈ x (MV6) ¬ ( ¬ x ⊕ y ) ⊕ y ≈ ¬ ( ¬ y ⊕ x ) ⊕ x . The variety MV of MV-algebras is coherent (see [24]), and generated as a quasivarietyby the positive universal class MV c of MV-algebras that are linearly ordered with respectto the defined lattice operations x ∧ y ≔ ¬ ( ¬ x ⊕¬ ( ¬ x ⊕ y )) and x ∨ y ≔ ¬ ( ¬ x ⊕ y ) ⊕ y . Again,it follows from Remark 2.2 that MV c has the equational variable projection property. ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS
Let us say that a class of algebras K has the conservative model extension property if for any finite set x , y and conjunctionof literals ψ ( x , y ), there exists a quantifier-free formula χ ( x ) satisfying(i) K | = ψ → χ (ii) for any A ∈ K generated by a ∈ A x such that A | = χ ( a ) and for any equation ε ( x ), K | = ψ + → ε = ⇒ A | = ε ( a ) , there exist an algebra B ∈ K extending A and b ∈ B such that B | = ψ ( a , b ).R emark ψ is satisfiable in K . Just observe that if this is not the case, then K | = ψ ↔ ⊥ andwe can let χ ≔ ⊥ . Moreover, if there is precisely one definable constant in K (whichis the case, e.g., for lattice-ordered abelian groups), we can assume that x is non-empty.To see this, suppose that x = ∅ and let χ ≔ ⊤ . Then (i) is clearly satisfied and for (ii),any A ∈ K generated by ∅ is trivial and satisfies all equations, so, since ψ is satisfiablein K , we can choose an algebra B ∈ K extending A and b ∈ B such that B | = ψ ( b ).In the case where K is a universal class of algebras admitting finite presentations(in particular, any quasivariety), the conservative model extension property is impliedby the conservative congruence extension property introduced by Wheeler in [34] (seeProposition A.3 of Appendix A). The following proposition, proved here for the sake ofcompleteness, is then a direct consequence of [34, Corollary 1, p. 319].P roposition roof . Let V be a locally finite variety and consider a finite set x , y and conjunctionof literals ψ ( x , y ). We can assume that ψ is satisfiable in V . Since V is locally finite,the free finitely generated algebra F V ( x ) is finite and has finitely many congruences θ , . . . , θ m . For each w ∈ F V ( x ) , choose an equation ε w such that b ε w = w . Now, foreach j ∈ { , . . . , m } , let ψ j ( x ) be the conjunction of literals in the set { ε w | w ∈ θ j }∪{¬ ε w | w < θ j } . Then, for any algebra A ∈ V generated by a tuple a ∈ A x , we have A | = ψ j ( a )if, and only if, the map F V ( x ) /θ j → A sending x j to a , where x j is the image of the tuple x under the composite map Tm ( x ) ։ F V ( x ) ։ F V ( x ) /θ j , is an isomorphism.Let S be the set of all j ∈ { , . . . , m } such that there exist B ∈ V extending F V ( x ) /θ j and b ∈ B satisfying B | = ψ ( x j , b ). Since ψ is satisfiable in V , there exist B ∈ V and( a , b ) ∈ B x , y such that B | = ψ ( a , b ). The following claim then implies that S , ∅ .C laim . Suppose that B ∈ V and B | = ψ ( a , b ) for some ( a , b ) ∈ B x , y . Then there exist j ∈ { , . . . , m } and an embedding F V ( x ) /θ j ֒ → B sending x j to a , and hence j ∈ S .P roof of C laim . By the homomorphism theorem for universal algebra (see, e.g., [7]),it su ffi ces to observe that the image of the homomorphism F V ( x ) → B sending theequivalence class of x to a is isomorphic to F V ( x ) /θ j for some j ∈ { , . . . , m } . ⊣ We now prove that the quantifier-free formula χ ( x ) ≔ g k ∈ S ψ k GEORGE METCALFE AND LUCA REGGIO satisfies conditions (i) and (ii) in the definition of the conservative model extensionproperty. For (i), we must show V | = ψ → χ . By the Claim, if B ∈ V and ( a , b ) ∈ B x , y satisfy B | = ψ ( a , b ), there exist j ∈ { , . . . , m } and an embedding F V ( x ) /θ j ֒ → B sending x j to a . So B | = ψ j ( a ), which implies B | = χ ( a ). For (ii), suppose that A ∈ V is generatedby a tuple a ∈ A x satisfying A | = χ ( a ). Let k ∈ S be such that A | = ψ k ( a ), and recall thatthere is an isomorphism F V ( x ) /θ k → A sending x k to a . By the definition of S , thereexist B ∈ V extending A and b ∈ B such that B | = ψ ( a , b ). ⊣ E xample LA c has the conservative modelextension property. Consider a finite set x , y and a conjunction of literals ψ ( x , y ). Wefirst assume that y appears in each literal of ψ and then settle the general case. For anytwo terms s , t , write s < t for the formula ( s ≤ t ) & ¬ ( s ≈ t ), where ≤ is the definablelattice order. In view of Remark 2.9, we can assume that x , ∅ . Moreover, without lossof generality (because the members of LA c are linearly ordered), we can assume that ψ = ψ g · · · g ψ m such that each disjunct is a conjunction of formulas of the form py ≤ t , py ≥ t , py < t , py > t , where t varies among the group terms in the variables x = x , . . . , x n , and p is a fixed non-zero natural number (just consider the least common multiple of the coe ffi cients of y involved). Let t j ( x ) , . . . , t j u ( x ) be the terms appearing in ψ j and let χ ( x ) be the formula & { g ≤ j ≤ m t j i △ t j k | j i , j k ∈ { j , . . . , j u } , △ ∈ { <, ≤} , and LA c | = ψ → g ≤ j ≤ m t j i △ t j k } & & { g i ∈ I ¬ ( x i ≈ | I ⊆ { , . . . , n } and LA c | = ψ → g i ∈ I ¬ ( x i ≈ } . Clearly, LA c | = ψ → χ , so condition (i) of the conservative model extension propertyis satisfied. For (ii), consider A ∈ LA c along with a tuple a ∈ A x such that A | = χ ( a ).If A is the one-element group and there is no B ∈ LA c satisfying (ii), then LA c | = ψ → g ni = ¬ ( x i ≈ LA c | = χ → g ni = ¬ ( x i ≈
0) by the definition of χ , contradicting thefact that A | = χ ( a ). If A is non-trivial, let B be the divisible hull of A and note that B is aninfinite member of LA c . We claim that there is a b ∈ B such that B | = ψ ( a , b ). If no such b exists, then for each 1 ≤ j ≤ m , a pair of inequations t j i ( a ) △ py and py △ t j k ( a ) of ψ j ( a , y )is unsatisfiable, for △ ∈ { <, ≤} . We settle the case where all these inequations are of theform t j i ( a ) < py and py < t j k ( a ), the other cases being very similar. Since B is divisibleand every divisible ordered abelian group is densely ordered, we get t j k ( a ) ≤ t j i ( a ).Moreover, t j i < py and py < t j k entail t j i < t j k , hence LA c | = χ → g ≤ j ≤ m t j i < t j k . Butthen A | = χ ( a ) implies t j i ( a ) < t j k ( a ) for some 1 ≤ j ≤ m , a contradiction.Finally, if we have a conjunction of literals of the form ψ ( x , y ) & ψ ′ ( x ), where ψ ′ isany conjunction of literals, the quantifier-free formula χ ∧ ψ ′ satisfies the conditions forthe conservative model extension property.E xample DL c of linear orders with endpoints, in thelanguage of bounded lattices, has the conservative model extension property. Consider afinite set x , y and a conjunction of literals ψ ( x , y ) satisfiable in DL c . Suppose that x = ∅ .If there is a non-trivial member of DL c satisfying ψ , then the formula χ ≔ ¬ (0 ≈ ψ is satisfied only by the trivial algebra, we can set χ ≔ ≈ x , ∅ . We assume that y appears in all literals of ψ (the general case thenfollows by reasoning as in Example 2.11). Using the fact that all members of DL c are ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS ψ is equivalent to a formula ψ g · · · g ψ m where eachdisjunct is a conjunction of formulas of the form y ≤ x , y ≥ x , y < x , y > x , where x varies among the variables x = x , . . . , x n . Assume first that the trivial algebrasatisfies ψ and let χ ( x ) be the formula & { g ( i , j ) ∈ J x i △ x j | J ⊆ { , . . . , n } , △ ∈ { <, ≤} , and DL c | = ψ → g ( i , j ) ∈ J x i △ x j } . Reasoning as in Example 2.11, it is not di ffi cult to see that χ satisfies the conditions forthe conservative model extension property. Just replace the divisible hull of an orderedabelian group with any dense linear order with endpoints which extends the given linearorder with endpoints A ∈ DL c (e.g., there is an embedding of A into the lexicographicproduct of A and [0 ,
1] which preserves the bounds).In the case where ψ is not satisfied in the trivial algebra, we replace χ by χ & ¬ (0 ≈ §3. Model completions. Let us first recall some relevant model-theoretic notions,referring to [8, Section 3.5] for further details. By a theory we will always mean afirst-order theory, i.e., a set of sentences over some first-order language L . We letTh( K ) denote the theory of a class K of L -structures, i.e., the set of L -sentences that aresatisfied by all members of K . Two theories T and T ′ are called co-theories if they entailthe same universal sentences. Semantically, T ′ is a co-theory of T if, and only if, everymodel of T embeds into a model of T ′ and vice versa. A theory T ∗ is model complete if every formula is equivalent over T ∗ to an existential formula; that is, model completetheories are those in which alternations of quantifiers can be eliminated. Semantically,a theory T ∗ is model complete if, and only if, every embedding between models of T ∗ is elementary. A theory T ∗ is a model companion of a theory T if it is a model completeco-theory of T . A model completion of a theory T is a model companion T ∗ of T suchthat for any model M of T , the theory of T ∗ together with the diagram of M is complete.Let us also recall that a class K of L -structures has the amalgamation property ifgiven any A , B , C ∈ K and embeddings f : A → B and g : A → C , there exist D ∈ K and embeddings h : B → D and k : C → D satisfying h f = kg .Below, we collect some useful facts related to model completions.P roposition ∀∃ -theory T has a model companion T ∗ , then T ∗ coincides with the theory ofthe existentially closed models of T .(c) If T ∗ is a model companion of a theory T , then T ∗ is a model completion of T if,and only if, the class of models of T has the amalgamation property.(d) A theory T has quantifier elimination if, and only if, T is model complete and theuniversal theory of T has the amalgamation property.These facts have the following immediate consequence, which we record here forfuture reference.P roposition T be a universal theory. A theory T ∗ is a model completion of T if, and only if, T ∗ is a co-theory of T that admits quantifier elimination. GEORGE METCALFE AND LUCA REGGIO
Our aim in this section is to prove the following characterization of universal classesof algebras whose theories admit a model completion.T heorem K be a universal class of algebras. Then the theory of K admits amodel completion if, and only if, K has the amalgamation property, variable projectionproperty, and conservative model extension property.For universal classes of algebras with finite presentations, this theorem specializesto [34, Theorem 5] (see Appendix A). Observe also that Theorem 3.3 combined withRemark 2.4 and Proposition 2.10 yields the following refinement in the case of locallyfinite varieties.C orollary V be a locally finite variety. Then thetheory of V admits a model completion if, and only if, V has the amalgamation property.We first settle the ‘if’ part of Theorem 3.3.P roposition K be a universal class of algebras that has the amalgamationproperty, variable projection property, and conservative model extension property. Thenthe theory of K has a model completion.P roof . Fix a countably infinite set of variables z and let J ≔ { ( ψ, y ) | ψ is a conjunction of literals with variables in z , and y ∈ z } . Consider j = ( ψ, y ) ∈ J . Let x be the set of variables occurring in ψ di ff erent from y .Since K has the variable projection property, there exists a quantifier-free formula ξ j ( x )such that K | = ψ + → ξ j and for every equation ε ( x ), K | = ψ + → ε = ⇒ K | = ξ j → ε. (1)Moreover, since K has the conservative model extension property, there is a quantifier-free formula χ j ( x ) satisfying the following conditions:(i) K | = ψ → χ j (ii) for any A ∈ K generated by a ∈ A x such that A | = χ j ( a ) and for any equation ε ( x ), K | = ψ + → ε = ⇒ A | = ε ( a ) , there exist an algebra B ∈ K extending A and b ∈ B such that B | = ψ ( a , b ).We define the first-order sentence τ j ≔ ∀ x [( ξ j & χ j ) → ∃ y .ψ ] . Let T ≔ Th( K ) be the theory of K and set T ∗ ≔ T ∪ { τ j | j ∈ J } . We claim that T ∗ isthe model completion of T . In view of Proposition 3.2, it su ffi ces to show that T ∗ hasquantifier elimination and is a co-theory of T . T ∗ has quantifier elimination. We prove that for any ( ψ, y ) ∈ J , T ⊢ ∀ x [( ∃ y .ψ ) → ( ξ j & χ j )] . (2)It follows then from the definition of T ∗ and (2) that T ∗ entails that any formula ∃ y .ψ ,where ψ is a conjunction of literals with variables in z and y ∈ z , is equivalent to aquantifier-free formula. So T ∗ has quantifier elimination (see, e.g., [33, Lemma 3.2.4]).For the proof of (2), fix an arbitrary j = ( ψ, y ) ∈ J . It su ffi ces to show that for anyalgebra A ∈ K and map g : x → A , A , g | = ∃ y .ψ = ⇒ A , g | = ξ j & χ j . ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS A , g | = ∃ y .ψ . Then A , f | = ψ for some map f : x , y → A extending g .Moreover, since K | = ψ + → ξ j and K | = ψ → χ j , it follows that A , f | = ξ j and A , f | = χ j .But ξ j and χ j have variables in x , so A , g | = ξ j and A , g | = χ j , yielding A , g | = ξ j & χ j . T ∗ is a co-theory of T . Since T ⊆ T ∗ , it su ffi ces to show that any universal sentenceentailed by T ∗ is entailed by T . First we show that for any j = ( ψ, y ) ∈ J , algebra A ∈ K ,and map g : x → A , A , g | = ξ j & χ j = ⇒ there exist B ∈ K and ι : A ֒ → B such that B , ι g | = ∃ y .ψ. (3)Suppose that A , g | = ξ j & χ j . We will assume first that the unique homomorphism˜ g : Tm ( x ) → A extending g is surjective, and hence that A is generated by the image a of x under g . Note that A | = χ j ( a ). Moreover, if ε ( x ) is any equation such that K | = ψ + → ε , then (1) yields K | = ξ j → ε and, since A | = ξ j ( a ), also A | = ε ( a ). Hence,by (ii), there exists an algebra B in K extending A and b ∈ B such that B | = ψ ( a , b ). So B , ι g | = ∃ y .ψ , where ι : A ֒ → B is the inclusion map.For the general case of (3), let A ′ be the image of Tm ( x ) under ˜ g in A . Since K isa universal class and A ′ embeds into A , also A ′ ∈ K . By the previous argument, thereexist B ′ ∈ K and ι ′ : A ′ ֒ → B ′ such that B ′ , ι ′ g | = ∃ y .ψ , witnessed by b ∈ B ′ , say.Since K has the amalgamation property, we obtain for the injection ι ′ and the inclusion A ′ ֒ → A , an extension ι : A ֒ → B with B ∈ K and an embedding λ : B ′ ֒ → B such thatthe following diagram commutes: A ′ B ′ A B ι ′ λι Since ψ is quantifier-free, we have B , ι g | = ∃ y .ψ witnessed by λ ( b ).To conclude the proof, let α be a universal sentence such that T ∗ ⊢ α . By thecompactness theorem of first-order logic, there exists a finite subset F ⊆ J such that T ∪ { τ j | j ∈ F } ⊢ α . We prove that, in fact, T ⊢ α . Consider any A ∈ K . Let w be theset of variables appearing in the scope of the universal quantifier in one of the sentences { τ j | j ∈ F } let and g : w → A be any map. By repeatedly applying (3), we obtain B ∈ K and ι : A ֒ → B such that B , ι g | = τ j for each j ∈ F . Now, since T ∪ { τ j | j ∈ F } ⊢ α ,we have B , ι g | = α . But A is a subalgebra of B and α is universal, so A , g | = α . Hence T ⊢ α as required. ⊣ E xample LA c of ordered abelian groups, definedover the algebraic language with operation symbols ∧ , ∨ , + , − ,
0, has the amalgamationproperty [30], variable projection property (see Example 2.7), and conservative modelextension property (see Example 2.11). Hence Proposition 3.5 yields the well-knownfact that the theory of LA c admits a model completion [32]. Similarly, the class MV c oflinearly ordered MV-algebras satisfies the conditions of Proposition 3.5 and its theorytherefore admits a model completion [26].E xample DL c of linear orders with endpoints, de-fined over the algebraic language of bounded lattices, has the amalgamation property, Note that although this class is often defined using other first-order languages (e.g., ≤ , + ), this di ff erenceis immaterial for determining the existence of a model completion. GEORGE METCALFE AND LUCA REGGIO variable projection property (see Example 2.5), and conservative model extension prop-erty (see Example 2.12). So Proposition 3.5 yields the well-known fact that the theoryof DL c has a model completion (see, e.g., [8]). Moreover, it follows easily from this re-sult that the theory of the universal class HA c of linearly ordered Heyting algebras has amodel completion. Note that any linear order with endpoints A ∈ DL c can be expandedto a linearly ordered Heyting algebra by defining a ⊃ b ≔ a ≤ b and a ⊃ b ≔ b otherwise. Indeed, the theory of HA c is a definitional extension of the theory of DL c (inthe sense of, e.g., [21]), where the binary function symbol ⊃ is defined by the formula ξ ( x , y , z ) ≔ ( x ≤ y & z ≈ g ( x > y & z ≈ y ) . Using the fact that ξ is quantifier-free, it is not di ffi cult to see that the existence of amodel completion for the theory of HA c follows directly from the existence of a modelcompletion for the theory of DL c .We now prove the ‘only if’ part of Theorem 3.3. Note first that if K is a universalclass of algebras with a model completion, then K has the amalgamation property byProposition 3.1.(c). We turn our attention next to the variable projection property.P roposition K has a model comple-tion, then K has the variable projection property.P roof . Let T ≔ Th( K ) be the theory of K , and let T ∗ be a model completion of T . Fixa finite set x , y and conjunction of equations ϕ ( x , y ). Since T ∗ has quantifier elimination,there exists a quantifier-free formula ξ ( x ) such that T ∗ ⊢ ∀ x [( ∃ y .ϕ ) ↔ ξ ] . (4)It follows from (4) that T ∗ ⊢ ∀ x , y ( ϕ → ξ ) and, since T and T ∗ have the same universalconsequences, T ⊢ ∀ x , y ( ϕ → ξ ). That is, K | = ϕ → ξ . It remains to show that for anyequation ε ( x ) satisfying K | = ϕ → ε , also K | = ξ → ε .Suppose that K = ξ → ε . Then there is an algebra A in K and a tuple a ∈ A x such that A | = ξ ( a ) and A = ε ( a ). Since T and T ∗ are co-theories and T is universal, there existsa model B of T ∗ extending A such that B ∈ K . Moreover, A | = ξ ( a ) implies B | = ξ ( a ),and hence B | = ∃ y .ϕ ( a , y ) by (4). Pick b ∈ B such that B | = ϕ ( a , b ). Since A = ε ( a ), weget B = ε ( a ). Hence K = ϕ → ε as required. ⊣ To complete the proof of Theorem 3.3, it remains to prove that whenever the theoryof a universal class of algebras has a model completion, this class has the conservativemodel extension property. In fact, we will show below that the existence of a modelcompletion is equivalent to a stronger property that directly implies the conservativemodel extension property. This stronger property is di ffi cult to check in concrete cases,but will be useful in Section 5 for establishing consequences of the existence of a modelcompletion for the definability of principal congruences.P roposition K be a universal class of algebras. The theory of K has a modelcompletion if, and only if, for any finite sets x , y and conjunction of literals ψ ( x , y ), thereis a quantifier-free formula χ ( x ) satisfying the following conditions:(i) K | = ψ → χ (ii) for any A ∈ K and tuple a ∈ A x satisfying A | = χ ( a ), there exist B ∈ K extending A and b ∈ B y such that B | = ψ ( a , b ). ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS K has a model completion, then K has the conservativemodel extension property.P roof . Let T ≔ Th( K ) be the theory of K , and suppose that T ∗ is a model completionof T . Fix finite sets x , y and a conjunction of literals ψ ( x , y ). Since T ∗ admits quantifierelimination, there exists a quantifier-free formula χ ( x ) such that T ∗ ⊢ ∀ x [( ∃ y .ψ ) ↔ χ ] . (5)It follows that T ∗ ⊢ ∀ x , y ( ψ → χ ), and since T and T ∗ are co-theories, T ⊢ ∀ x , y ( ψ → χ ) . (6)We show now that χ satisfies (i) and (ii).Condition (i) is an immediate consequence of (6). For (ii), fix an algebra A ∈ K anda tuple a ∈ A x such that A | = χ ( a ). We will exhibit B in K which extends A , and a tuple b ∈ B y such that B | = ψ ( a , b ). Since T and T ∗ are co-theories and T is universal, thereexists a model B of T ∗ extending A such that B ∈ K . Since B | = χ ( a ), it follows by (5)that there is a tuple b ∈ B y such that B | = ψ ( a , b ). This settles (ii).For the converse direction, we only sketch the proof, as it is an easy modification ofthe proof of Proposition 3.5. As before, we consider the theory T ∗ ≔ T ∪ { τ j | j ∈ J } ,but in this case τ j is defined as ∀ x ( χ j → ∃ y .ψ ), where the quantifier-free formula χ j satisfies (i) and (ii). To show that T ∗ has quantifier elimination, it su ffi ces to show that T ⊢ ∀ x ( ∃ y .ψ → χ j ) for all j = ( ψ, y ) ∈ J , which follows by (i) (cf. the proof ofProposition 3.5). On the other hand, (ii) yields the following property similar to (3): if A , g | = χ j , then there exist B ∈ K and ι : A ֒ → B such that B , ι g | = ∃ y .ψ . Reasoning as inthe last part of the proof of Proposition 3.5, we conclude that T and T ∗ are co-theories.It follows then by Proposition 3.2 that T ∗ is the model completion of T . ⊣ §4. Compact congruences. In this section, we show that for varieties of algebrassatisfying certain congruence lattice conditions, the rather complicated conservativemodel extension property appearing in Theorem 3.3 can be replaced by a more amenableequational variable restriction property. The resulting characterization is a slightly moregeneral version of a theorem of Ghilardi and Zawadowski [16, Theorem 4].Recall that the compact (equivalently, finitely generated) congruences of an algebra A ordered by set-theoretic inclusion always form a join-semilattice. A class of algebras K is said to have the compact intersection property if the join-semilattice of compactcongruences of each A ∈ K forms a lattice, i.e., the intersection of any two compactcongruences of A is compact. Recall also that K is said to be congruence distributive ifthe congruence lattice of each A ∈ K is distributive. The following lemma describes auseful syntactic consequence of these two properties.L emma V be a congruence distributive variety that has the compact intersec-tion property. For any finite sets x , y and conjunctions of equations ϕ ( x ) , ϕ ( x ), thereexists a conjunction of equations π ( x ) such that for any equation ε ( x , y ), V | = ( ϕ g ϕ ) → ε ⇐⇒ V | = π → ε. P roof . Let x be any finite set and let ϕ ( x ) and ϕ ( x ) be conjunctions of equations.We let z be any countably infinite set with x ∩ z = ∅ and define θ i ≔ Cg F V ( x , z ) ( { b σ | σ is an equation of ϕ i } ) for i ∈ { , } . Since θ and θ are compact, so, by assumption,2 GEORGE METCALFE AND LUCA REGGIO is their intersection; that is, there exist a finite set w ⊆ z and a conjunction of equations σ ( x , w ) , . . . , σ n ( x , w ) such that { b σ , . . . , b σ n } generates θ ∩ θ . Let f : x , z → F V ( x , z )be a map sending each x ∈ x to x and each w ∈ w to some u ∈ F V ( x ), such that thegenerators x , z of F V ( x , z ) are contained in the image of f . Clearly, the unique homo-morphism g : F V ( x , z ) → F V ( x , z ) extending f is surjective. Define g ∗ : Con F V ( x , z ) → Con F V ( x , z ) by g ∗ ( θ ) ≔ Cg F V ( x , z ) ( { ( g ( a ) , g ( b )) | ( a , b ) ∈ θ } ). Since V is congruence dis-tributive and g is surjective, g ∗ is a lattice homomorphism (cf. [2, Lemma 1.11]) and,in particular, g ∗ ( θ ∩ θ ) = g ∗ ( θ ) ∩ g ∗ ( θ ) = θ ∩ θ . Hence also { g ( b σ ) , . . . , g ( b σ n ) } generates θ ∩ θ , where g ( b σ j ) ≔ ( g ( s ) , g ( t )) whenever b σ j = ( s , t ), and we let π ( x ) bethe conjunction of the equations σ ( x , u , . . . , u ) , . . . , σ n ( x , u , . . . , u ).For any finite set y and equation ε ( x , y ), we may assume without loss of generalitythat y ⊆ z and use Lemma 2.1 to obtain V | = ( ϕ g ϕ ) → ε ⇐⇒ V | = ϕ → ε and V | = ϕ → ε ⇐⇒ b ε ∈ θ and b ε ∈ θ ⇐⇒ b ε ∈ θ ∩ θ ⇐⇒ V | = π → ε. ⊣ R emark K has first-order definable principal congruences if there exists a formula α ( x , x , y , y ) satisfying for all A ∈ K and a , a , b , b ∈ A ,( a , a ) ∈ Cg A ( b , b ) ⇐⇒ A | = α ( a , a , b , b ) , where Cg A ( b , b ) ≔ Cg A ( { ( b , b ) } ) is the smallest congruence of A containing the pair( b , b ). If α can be chosen to be a quantifier-free formula or conjunction of equations,then K is said to have, respectively, quantifier-free definable principal congruences or equationally definable principal congruences . For a variety V , it is known that havingequationally definable principal congruences corresponds to a property of the associatedconsequence relation of V (referred to as a deduction theorem) and a property of thejoin-semilattices of compact congruences of members of V .P roposition V :(1) V has equationally definable principal congruences.(2) There exists a conjunction of equations ϕ ( x , x , y , y ) such that for any conjunc-tion of equations π and terms s , s , t , t , V | = ( π & ( s ≈ s )) → ( t ≈ t ) ⇐⇒ V | = π → ϕ ( s , s , t , t ) . (3) The join-semilattice of compact congruences of each A ∈ V is dually Brouwerian,i.e., for any compact congruences θ , θ of A , there exists a compact congruence θ − θ of A such that for any compact congruence θ of A , θ − θ ⊆ θ ⇐⇒ θ ⊆ θ ∨ θ . A variety that has equationally definable principal congruences has two further usefulproperties. Recall that a variety V has the congruence extension property if for all A ∈ V ,any congruence of a subalgebra of A extends to a congruence of A . ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS roposition V has the compact intersection property and equationallydefinable principal congruences if, and only if, the compact congruences of each A ∈ V ordered by set-theoretic inclusion form a dually Brouwerian lattice.Let us say now that a class of algebras K has the equational variable restrictionproperty if for any finite set x , y and conjunction of equations γ ( x , y ), there exists aformula π ( x ) that is either ⊥ or a conjunction of equations such that K | = π → γ and forany conjunction of equations ϕ ( x ), K | = ϕ → γ = ⇒ K | = ϕ → π. The main result of this section is the following characterization theorem.T heorem V be a variety that has the compact intersection property and equa-tionally definable principal congruences. Then the theory of V admits a model com-pletion if, and only if, V is coherent and has the amalgamation property and equationalvariable restriction property.We show first that the conditions in Theorem 4.5 are su ffi cient even in the absence ofthe compact intersection property.P roposition V be a variety that has equationally definable principal congru-ences. If V is coherent and has the amalgamation property and the equational variablerestriction property, then the theory of V admits a model completion.P roof . By Corollary 3.4, it is enough to prove that V has the conservative modelextension property. Consider a finite set x , y and conjunction of literals ψ ( x , y ), assumingwithout loss of generality that ψ is satisfiable in V (cf. Remark 2.9). By coherence, thereexists a conjunction of equations ϕ ( x ) such that V | = ψ + → ϕ and for any equation ε ( x ), V | = ψ + → ε = ⇒ V | = ϕ → ε. Let σ , . . . , σ m be the equations of ψ − . Since V has equationally definable principalcongruences, it follows by Proposition 4.3 that for each i ∈ { , . . . , m } , there exists aconjunction of equations π i ( x , y ) such that for any conjunction of equations γ ( x , y ), V | = ( γ & ψ + ) → σ i ⇐⇒ V | = γ → π i . (7)The equational variable restriction property yields a formula π ′ i ( x ) for each i ∈ { , . . . , m } that is either ⊥ or a conjunction of equations such that V | = π ′ i → π i and for anyconjunction of equations ϕ ′ ( x ), V | = ϕ ′ → π i = ⇒ V | = ϕ ′ → π ′ i . (8)We claim that the quantifier-free formula χ ≔ ϕ & ¬ π ′ & · · · & ¬ π ′ m satisfies the conditionsin the definition of conservative model extension property.Note first that, since V | = ψ → ψ + and V | = ψ + → ϕ , also V | = ψ → ϕ . To concludethat V | = ψ → χ , it remains to show that V | = ψ → ¬ π ′ i for each i ∈ { , . . . , m } . Let i ∈ { , . . . , m } . Since V | = π ′ i → π i and V | = ( π i & ψ + ) → σ i , also V | = ( π ′ i & ψ + ) → σ i .But then V | = ( ¬ σ i & ψ + ) → ¬ π ′ i and hence V | = ψ → ¬ π ′ i as required.4 GEORGE METCALFE AND LUCA REGGIO
Now, consider an algebra A ∈ V generated by a ∈ A x such that A | = χ ( a ) and for anyequation ε ( x ), V | = ψ + → ε = ⇒ A | = ε ( a ) . (9)Let f : F V ( x ) ։ A be the surjective homomorphism mapping the generators of F V ( x )to the elements of a . We identify F V ( x ) with a subalgebra of F V ( x , y ) and set θ ≔ Cg F V ( x , y ) (ker f ∪ θ ′ ), where θ ′ ≔ Cg F V ( x , y ) ( { b σ | σ is an equation of ψ + } ). Let B ≔ F V ( x , y ) /θ and let g be the natural homomorphism from F V ( x , y ) onto B . Trivially,ker f ⊆ θ ∩ F V ( x ) , and hence the inclusion F V ( x ) ֒ → F V ( x , y ) yields a homomorphism A → B sending f ( x ) to g ( x ) for each x ∈ x , as illustrated by the following diagram: F V ( x ) F V ( x , y ) A B f g
To prove that this homomorphism is an embedding, it su ffi ces to show that θ ∩ F V ( x ) ⊆ ker f . Note first that V has the congruence extension property and hence, by [27, Propo-sition 16], we have θ ∩ F V ( x ) = ker f ∨ ( θ ′ ∩ F V ( x ) ). It therefore su ffi ces to provethat θ ′ ∩ F V ( x ) ⊆ ker f . Consider any equation ε ( x ) such that b ε ∈ θ ′ ∩ F V ( x ) . Anapplication of Lemma 2.1 yields V | = ψ + → ε and, by (9), we obtain A | = ε ( a ). Hence b ε ∈ ker f as required.To conclude the proof, we show that B | = ψ ( a , b ), where b is the image under g ofthe equivalence class of y . Since θ ′ ⊆ θ = ker g , we have B | = ψ + ( a , b ). Supposefor a contradiction that B | = σ i ( a , b ) for some i ∈ { , . . . , m } . Then b σ i ∈ θ , and so V | = ( ψ + & γ ) → σ i for some conjunction γ ( x ) of equations in ker f . By (7), we have V | = γ → π i and hence, by (8), also V | = γ → π ′ i . Together with A | = γ ( a ), this entails A | = π ′ i ( a ), contradicting the fact that A | = χ ( a ). ⊣ We now complete the proof of Theorem 4.5 by establishing the necessity of the statedconditions, recalling that a variety that has equationally definable principal congruencesis congruence distributive (Proposition 4.4).P roposition V be a congruence distributive variety that has the compact in-tersection property. If the theory of V admits a model completion, then V is coherentand has the amalgamation property and equational variable restriction property.P roof . If the theory of V has a model completion, then V is coherent and has theamalgamation property by Corollary 3.4. Hence, it remains to settle the equationalvariable restriction property. To this end, consider a finite set x , y and conjunction ofequations γ ( x , y ). We must find a formula π ( x ) that is either ⊥ or a conjunction ofequations such that V | = π → γ and for any conjunction of equations ϕ ( x ), V | = ϕ → γ = ⇒ V | = ϕ → π. Let T ≔ Th( V ) be the theory of V , and let T ∗ be a model completion of T . Since T ∗ hasquantifier elimination, there exists a quantifier-free formula ξ ( x ) satisfying T ∗ ⊢ ∀ x [( ∀ y .γ ) ↔ ξ ] . So T ∗ ⊢ ∀ x , y ( ξ → γ ) and, since T and T ∗ are co-theories, T ⊢ ∀ x , y ( ξ → γ ). Assumewithout loss of generality that ξ = ξ g · · · g ξ m where each ξ i is a conjunction of literals. ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS i ∈ { , . . . , m } , we have V | = ξ i → γ and so, by the disjunction property forvarieties, either V | = ξ + i → γ or V | = ¬ ξ i .Now let J ≔ { i ∈ { , . . . , m } | ξ i is satisfiable in V } . Then V | = g i ∈ J ξ + i → γ , notingthat g i ∈ J ξ + i = ⊥ for J = ∅ . There exists a formula π ( x ) that is either ⊥ (if J = ∅ ) or, byLemma 4.1, a conjunction of equations such that for any finite set z and equation ε ( x , z ), V | = g i ∈ J ξ + i → ε ⇐⇒ V | = π → ε. In particular, V | = π → γ . Now let ϕ ( x ) be any conjunction of equations such that V | = ϕ → γ . Then T ⊢ ∀ x [ ϕ → ( ∀ y .γ )] and, since T and T ∗ are co-theories, T ∗ ⊢∀ x [ ϕ → ( ∀ y .γ )]. Hence T ∗ ⊢ ∀ x ( ϕ → ξ ) and, again using the fact that T and T ∗ areco-theories, T ⊢ ∀ x ( ϕ → ξ ), yielding V | = ϕ → ξ . But then V | = ϕ → g i ∈ J ξ + i and,since V | = g i ∈ J ξ + i → π , by the above equivalence, V | = ϕ → π as required. ⊣ R emark A ∈ V has a bottom element, or, equivalently, that there isa variable-free conjunction of equations π such that V | = π → ε for every equation ε .However, this condition is not satisfied for all the varieties of interest in this paper suchas the variety of lattice-ordered abelian groups.Finally, in this section, we show that for varieties that have the congruence extensionproperty, a small generalization of the equational variable restriction property impliesthe amalgamation property. Let us first recall the following well-known relationshipbetween amalgamation and deductive interpolation.P roposition V be a variety that has the congruenceextension property. Then V has the amalgamation property if, and only if, it has the deductive interpolation property ; that is, for any finite sets x , y , z and conjunctions ofequations ϕ ( x , y ), γ ( x , z ) satisfying V | = ϕ → γ , there exists a conjunction of equations π ( x ) such that V | = ϕ → π and V | = π → γ .It follows easily that the amalgamation property and equational variable restrictionproperty can be combined into a single syntactic uniform interpolation property.C orollary V be a variety that has the congruence extension property. Thenthe following statements are equivalent:(1) V has the amalgamation property and the equational variable restriction property.(2) For any finite set x , y and conjunction of equations γ ( x , y ), there exists a formula π ( x ) that is either ⊥ or a conjunction of equations such that V | = π → γ and forany conjunction of equations ϕ ( x , z ), V | = ϕ → γ = ⇒ V | = ϕ → π. P roof . (1) ⇒ (2) Suppose that V has the amalgamation property and the equationalvariable restriction property, and consider a finite set x , y and conjunction of equations γ ( x , y ). By the equational variable restriction property, there exists a formula π ( x ) thatis either ⊥ or a conjunction of equations such that V | = π → γ and for any conjunctionof equations ϕ ( x ), V | = ϕ → γ = ⇒ V | = ϕ → π. GEORGE METCALFE AND LUCA REGGIO
Now consider a conjunction of equations ϕ ( x , z ) satisfying V | = ϕ → γ . Since V hasthe congruence extension property and the amalgamation property, by Proposition 4.9,there exists a conjunction of equations π ′ ( x ) such that V | = ϕ → π ′ and V | = π ′ → γ . Bythe above implication, V | = π ′ → π and hence also V | = ϕ → π as required.(2) ⇒ (1) The amalgamation property and the equational variable restriction propertyboth follow directly from condition (2) and Proposition 4.9. ⊣ §5. Parametrically definable principal congruences. In this section, we show thatif a variety of algebras satisfies certain congruence lattice conditions and has a modelcompletion, then it must have equationally definable principal congruences. Our maintechnical tool is a weaker condition for defining principal congruences inspired by thework of Glass and Pierce on existentially complete lattice-ordered abelian groups [19].We say that a class of algebras K has parametrically definable principal congruences if there exists a quantifier-free formula ξ ( x , x , y , y , z ) such that for each A ∈ K andall a , a , b , b ∈ A ,( a , a ) ∈ Cg A ( b , b ) ⇐⇒ any B ∈ K extending A satisfies B | = ∀ z .ξ ( a , a , b , b , z ) . Clearly, if a class of algebras has equationally definable principal congruences, it hasparametrically definable principal congruences. However, there are classes of algebrasthat have parametrically definable principal congruences that do not have even first-order definable principal congruences (see Example 5.8). The following propositionshows that this is not the case for a universal class of algebras whose theory admits amodel completion.P roposition K be a universal class of algebras with parametrically definableprincipal congruences. If the theory of K has a model completion, then K has quantifier-free definable principal congruences.P roof . Suppose that K has parametrically definable principal congruences, witnessedby a quantifier-free formula ξ ( x , x , y , y , z ), and that the theory of K has a modelcompletion. We can assume that ¬ ξ = ψ g · · · g ψ n , where each ψ i is a conjunction ofliterals. By Proposition 3.9, for each i ∈ { , . . . , n } , there exists a quantifier-free formula χ i ( x , x , y , y ) such that(i) K | = ψ i → χ i (ii) for any A ∈ K , and a , a , b , b ∈ A satisfying A | = χ i ( a , a , b , b ), there existsan extension B ∈ K of A such that B | = ∃ z .ψ i ( a , a , b , b , z ).We claim that for each A ∈ K and all a , a , b , b ∈ A ,( a , a ) ∈ Cg A ( b , b ) ⇐⇒ A | = & ≤ i ≤ n ¬ χ i ( a , a , b , b ) , and hence that K has quantifier-free definable principal congruences. Suppose firstthat ( a , a ) ∈ Cg A ( b , b ). By assumption, any B ∈ K extending A satisfies B | = ∀ z .ξ ( a , a , b , b , z ) and hence B = ∃ z .ψ i ( a , a , b , b , z ) for each i ∈ { , . . . , n } . Butthen, by (ii), also A | = ¬ χ i ( a , a , b , b ) for each i ∈ { , . . . , n } . Now suppose that( a , a ) < Cg A ( b , b ). By assumption, there exists B ∈ K extending A such that B = ∀ z .ξ ( a , a , b , b , z ), and hence B | = ∃ z .ψ i ( a , a , b , b , z ) for some i ∈ { , . . . , n } .But then, since K | = ψ i → χ i by (i), also B | = χ i ( a , a , b , b ). Since χ i is quantifier-free,it follows that A | = χ i ( a , a , b , b ). ⊣ ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS xample α ( x , y ) such that for any abelian group G , an element a ∈ G belongs to thesubgroup of G generated by an element b ∈ G if, and only if, G | = α ( a , b ). But thenthe sentence ∃ y ∀ x .α ( x , y ) would define the class of cyclic groups, contradicting the factthat this class is not elementary. On the other hand, the theory of abelian groups doesadmit a model completion (cf. [10]). Proposition 5.1 therefore tells us that the varietyof abelian groups does not have parametrically definable principal congruences.We strengthen Proposition 5.1 by exploiting the following fact, due to Fried, Gr¨atzer,and Quackenbush [11].P roposition V be a congruence distributive va-riety that has the congruence extension property. Then V has first-order definable prin-cipal congruences if, and only if, it has equationally definable principal congruences.Combining this fact with Proposition 5.1 yields the following result.P roposition V be a congruence distributive variety that has the congruenceextension property and parametrically definable principal congruences. If the theory of V has a model completion, then V has equationally definable principal congruences.We now provide a su ffi cient condition for a variety with the congruence extensionproperty to have parametrically definable principal congruences, expressed as a propertyof the associated consequence relation that is often more straightforward to establish.Let us say that a variety V has a guarded deduction theorem if there exist conjunctionsof equations γ ( x , x , y , y , z ), ϕ ( x , x , y , y , z ) such that for every finite set w with w ∩ z = ∅ , conjunction of equations π ( w ), and terms s ( w ) , s ( w ) , t ( w ) , t ( w ),(i) V | = ( π & ( t ≈ t )) → ( s ≈ s ) ⇐⇒ V | = π → ∀ z . ( γ → ϕ )( s , s , t , t , z )(ii) V | = ( π & γ ( s , s , t , t , z )) → σ ⇐⇒ V | = π → σ for any equation σ ( w ).By Theorem 4.3, every variety that has equationally definable principal congruenceshas a guarded deduction theorem. Moreover, this property has the following algebraiccharacterization:P roposition V :(1) V has a guarded deduction theorem.(2) There exist conjunctions of equations γ ( x , x , y , y , z ) , ϕ ( x , x , y , y , z ) such thatfor any finitely generated algebra A ∈ V and a , a , b , b ∈ A , there exists anembedding of A into some member of V satisfying ∃ z .γ ( a , a , b , b , z ), and( a , a ) ∈ Cg A ( b , b ) ⇐⇒ B | = ∀ z . ( γ → ϕ )( h ( a ) , h ( a ) , h ( b ) , h ( b ) , z )for any B ∈ V and homomorphism h : A → B .P roof . (1) ⇒ (2) Let γ ( x , x , y , y , z ) and ϕ ( x , x , y , y , z ) be conjunctions ofequations satisfying the conditions for a guarded deduction theorem, and fix a finitelygenerated algebra A ∈ V and a , a , b , b ∈ A . We claim that γ and ϕ satisfy theequivalence given in (2).Note first that since A is finitely generated, there exists a surjective homomorphism f : Tm ( w ) ։ A for some finite set w with w ∩ z = ∅ . Let s , s , t , t be terms in thepreimages under f of a , a , b , b , respectively. Now consider ( a , a ) ∈ Cg A ( b , b ).8 GEORGE METCALFE AND LUCA REGGIO
Since A is isomorphic to Tm ( w ) / ker f , by the correspondence theorem for universalalgebra, ( s , s ) ∈ ker f ∨ Cg Tm ( w ) ( t , t ). Moreover, since any congruence is the directedunion of the compact congruences below it, there exists a finite subset S ⊆ ker f suchthat ( s , s ) ∈ Cg Tm ( w ) ( S ) ∨ Cg Tm ( w ) ( t , t ). Let π ( w ) be the conjunction of the equationsin S . It follows easily from Lemma 2.1 that V | = ( π & ( t ≈ t )) → ( s ≈ s ). Hence V | = π → ∀ z . ( γ → ϕ )( s , s , t , t , z ), by condition (i) in the definition of a guardeddeduction theorem. But A | = π ( f ( w )), so for any B ∈ V and homomorphism h : A → B , B | = ∀ z . ( γ → ϕ )( h ( a ) , h ( a ) , h ( b ) , h ( b ) , z ) . Now suppose that B | = ∀ z . ( γ → ϕ )( h ( a ) , h ( a ) , h ( b ) , h ( b ) , z ) for any B ∈ V andhomomorphism h : A → B . Let D + ( A ) be the positive diagram of A , i.e., the set of allatomic sentences in the language extended with names for the elements of A that aresatisfied in A . Then Th( V ) ∪ D + ( A ) ⊢ ∀ z . ( γ → ϕ )( a , a , b , b , z ) , and hence, by the compactness theorem of first-order logic, there exists a finite subset Σ ⊆ D + ( A ) such that Th( V ) ∪ Σ ⊢ ∀ z . ( γ → ϕ )( a , a , b , b , z ). For each member of Σ ,consider an equation in its preimage under the surjection Tm ( w ) ։ A . This yields afinite subset of ker f , and letting π ( w ) denote the conjunction of the equations in this set,we obtain V | = π → ∀ z . ( γ → ϕ )( s , s , t , t , z ). By condition (i) in the definition of aguarded deduction theorem, we obtain V | = ( π &( t ≈ t )) → ( s ≈ s ). Now let q : A ։ A / Cg A ( b , b ) be the natural quotient map. Since A / Cg A ( b , b ) , q f | = π & ( t ≈ t ), wehave A / Cg A ( b , b ) , q f | = s ≈ s , i.e., ( a , a ) ∈ Cg A ( b , b ).To conclude, it remains to show that A embeds into some member of V satisfying ∃ z .γ ( a , a , b , b , z ). Let B be the quotient of F V ( w , z ) with respect to the congruenceCg F V ( w , z ) ( { b ε | ε ∈ ker f } ∪ { b σ | σ is an equation of γ ( s , s , t , t , z ) } ). Then B satisfies ∃ z .γ ( a , a , b , b , z ) and it is not di ffi cult to see, using condition (ii) in the definition ofa guarded deduction theorem, that B extends A .(2) ⇒ (1) Let γ ( x , x , y , y , z ) and ϕ ( x , x , y , y , z ) be conjunctions of equationssatisfying the conditions in (2). Consider a finite set w disjoint from z , a conjunction ofequations π ( w ), and terms s ( w ) , s ( w ) , t ( w ) , t ( w ).Let A be the quotient of F V ( w ) with respect to Cg F V ( w ) ( { b ε | ε is an equation of π } ),with quotient map f : F V ( w ) ։ A . Let q : Tm ( w ) ։ F V ( w ) denote the natural quotientmap and define a ≔ f ( q ( s )), a ≔ f ( q ( s )), b ≔ f ( q ( t )), and b ≔ f ( q ( t )). ByLemma 2.1, ( a , a ) ∈ Cg A ( b , b ) if, and only if, V | = ( π & ( t ≈ t )) → ( s ≈ s ).Hence, in order to settle condition (i) in the definition of a guarded deduction theorem,it remains to show that V | = π → ∀ z . ( γ → ϕ )( s , s , t , t , z ) if, and only if, for any B ∈ V and homomorphism h : A → B , B | = ∀ z . ( γ → ϕ )( h ( a ) , h ( a ) , h ( b ) , h ( b ) , z ) . This follows by reasoning as in the proof of (1) ⇒ (2).With respect to condition (ii) in the definition of a guarded deduction theorem, let B denote the quotient of F V ( w , z ) with respect toCg F V ( w , z ) ( { b ε | ε is an equation of π ( w ) } ∪ { b σ | σ is an equation of γ ( s , s , t , t , z ) } ) . Note that there is a canonical homomorphism k : A → B . By assumption, there is anembedding j : A ֒ → B ′ with B ′ ∈ V and B ′ | = ∃ z .γ ( a , a , b , b , z ). It follows easilythat there is a homomorphism h : B → B ′ such that j = hk . Since j is injective, so ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS k . Hence, by Lemma 2.1, we have that V | = ( π & γ ( s , s , t , t , z )) → σ ′ implies V | = π → σ ′ for any equation σ ′ ( w ). ⊣ P roposition V be a variety that has the congruence extension property and aguarded deduction theorem. Then V has parametrically definable principal congruences.P roof . It su ffi ces to show that V has parametrically definable principal congruenceswhenever the property in condition (2) of Proposition 5.5 holds. Let γ and ϕ be as inthe latter property and set ξ ( x , x , y , y , z ) ≔ γ → ϕ .We claim that, for every A ∈ V and a , a , b , b ∈ A ,( a , a ) ∈ Cg A ( b , b ) ⇐⇒ each B ∈ V extending A satisfies B | = ∀ z .ξ ( a , a , b , b , z ) . Suppose first that ( a , a ) ∈ Cg A ( b , b ). By the congruence extension property, ( a , a ) ∈ Cg A ′ ( b , b ), where A ′ is the subalgebra of A generated by a , a , b , b . Since A ′ is afinitely generated member of V , it follows from condition (2) of Proposition 5.5 that B | = ∀ z .ξ ( a , a , b , b , z ) for every B ∈ V extending A ′ , and hence, in particular, forevery B ∈ V extending A .Now suppose that ( a , a ) < Cg A ( b , b ). We assume first that A is finitely generated,and then deduce the general case. Let w be a finite set such that w ∩ z = ∅ and thereexists a surjective homomorphism f : Tm ( w ) ։ A . We can assume without loss ofgenerality that x , x , y , y ∈ w , and also f ( x ) = b , f ( x ) = b , f ( y ) = a , f ( y ) = a .Let B ≔ F V ( w , z ) /θ , where θ ≔ Cg F V ( w , z ) ( { b ε | ε ∈ ker f } ∪ { b σ | σ is an equation of γ } ) . We claim that B is an extension of A . Note that there is a canonical homomorphism j : A → B sending a , a , b , b to the equivalence classes of x , x , y , y , respec-tively. By assumption, there is an embedding i : A → B ′ with B ′ ∈ V and B ′ | = ∃ z .γ ( a , a , b , b , z ). It is not di ffi cult to see that i factors through j . Since the for-mer is injective, so is the latter.Now, by assumption, there exist C ∈ V and a homomorphism h : A → C satisfying C | = ∃ z . ( γ & ¬ ϕ i )( h ( a ) , h ( a ) , h ( b ) , h ( b ) , z ) for some equation ϕ i of ϕ . It followsthat there is a homomorphism k : Tm ( w , z ) → C which factors through the composite g : Tm ( w , z ) ։ F V ( w , z ) ։ B and satisfies ϕ i < ker k . So ϕ i < ker g , showing that B isan extension of A such that B = ∀ z .ξ ( a , a , b , b , z ).To conclude, consider any A ∈ V and suppose that B | = ∀ z .ξ ( a , a , b , b , z ) for all B ∈ V extending A . Then Th( V ) ∪ D ( A ) ⊢ ∀ z .ξ ( a , a , b , b , z ), where D ( A ) is thediagram of A , i.e., the collection of all atomic sentences and negated atomic sentencesin the language extended with names for the elements of A that are satisfied in A . Bythe compactness theorem of first-order logic, there is a finite subset Σ ⊆ D ( A ) suchthat Th( V ) ∪ Σ ⊢ ∀ z .ξ ( a , a , b , b , z ). Let A ′ be the subalgebra of A generated by a , a , b , b and the elements named by Σ . As the diagram of A ′ contains Σ , every B ′ ∈ V extending A ′ satisfies B ′ | = ∀ z .ξ ( a , a , b , b , z ). Note that A ′ is a finitely generatedmember of V and so, by the argument above, ( a , a ) ∈ Cg A ′ ( b , b ) ⊆ Cg A ( b , b ). ⊣ Combining Propositions 5.4 and 5.6 yields the main result of this section.T heorem V be a congruence distributive variety that has the congruence ex-tension property and a guarded deduction theorem. If the theory of V has a modelcompletion, then V has equationally definable principal congruences.0 GEORGE METCALFE AND LUCA REGGIO E xample LA is congruence-distributive and has the congruence extension property, but does not have equationally(or even first-order) definable principal congruences. Hence it su ffi ces, by Theorem 5.7,to observe that the formulas γ ≔ ( x − x ) ∧ ( x − x ) ∧ ≤ z & (( y − y ) ∧ ( y − y ) ∧ ∨ z ≈ ϕ ≔ z ≈ LA . §6. Varieties of pointed residuated lattices. In this section, we apply the resultsfrom the previous sections to a class of algebras that provide algebraic semantics forsubstructural logics and includes (up to term-equivalence) all lattice-ordered groups,MV-algebras, and Heyting algebras (see, e.g., [5, 13, 28]).A pointed residuated lattice is an algebra A = h A , ∧ , ∨ , · , \ , /, e , i such that h A , ∧ , ∨i is a lattice, h A , · , e i is a monoid, and \ , / are left and right residuals, respectively, of · inthe underlying lattice order, i.e., for all a , b , c ∈ A , b ≤ a \ c ⇐⇒ a · b ≤ c ⇐⇒ a ≤ c / b . It will be useful to define a further binary operation symbol x ≡ y ≔ ( x \ y ) ∧ ( y \ x ) ∧ e,noting that ( a ≡ b ) = e for a , b ∈ A if, and only if, a = b . We also define a ≔ e and a n + ≔ a · a n for any a ∈ A and all n ∈ N .Pointed residuated lattices form a congruence distributive variety [5], and include as anotable subvariety, the class CPRL of commutative pointed residuated lattices satisfying x · y ≈ y · x . In particular, a Heyting algebra is term-equivalent to a commutative pointedresiduated lattice satisfying x · y ≈ x ∧ y and x ∧ ≈
0, and a Boolean algebra isterm-equivalent to a Heyting algebra satisfying ( x \ \ ≈ x .Let V be a variety of pointed residuated lattices. We denote the class of linearlyordered members of V by V c , and call V semilinear if V = ISP ( V c ). Semilinearity canbe expressed equationally; in particular, a variety of commutative pointed residuatedlattices is semilinear if, and only if, it satisfies e ≈ (( x \ y ) ∧ e) ∨ (( y \ x ) ∧ e) [5].E xample LA of lattice-ordered abelian groups is term-equivalent tothe variety of commutative pointed residuated lattices satisfying e ≈ x · ( x \ e) ≈ e.More precisely, if h L , ∧ , ∨ , + , − , i ∈ LA and we define x · y ≔ x + y , x \ y ≔ y − x ,and x / y ≔ x − y , then h L , ∧ , ∨ , · , \ , /, , i ∈ CPRL satisfies e ≈ x · ( x \ e) ≈ e.Conversely, if L ∈ CPRL satisfies e ≈ x · ( x \ e) ≈ e and we define x + y ≔ x · y and − x ≔ x \ e, then h L , ∧ , ∨ , + , − , i ∈ LA . Recall also that LA = ISP ( LA c ), so thecorresponding variety of pointed residuated lattices is semilinear.E xample MV of MV-algebras is term-equivalent to the variety ofcommutative pointed residuated lattices satisfying x ∨ y ≈ ( x \ y ) \ y and x ∧ ≈
0. Moreprecisely, if h M , ⊕ , ¬ , i ∈ MV and we define x · y ≔ ¬ ( ¬ x ⊕ ¬ y ), x \ y ≔ ¬ x ⊕ y , x / y ≔ x ⊕ ¬ y , and e ≔ ¬
0, then h M , ∧ , ∨ , · , \ , /, e , i ∈ CPRL satisfies x ∨ y ≈ ( x \ y ) \ y and x ∧ ≈
0. Conversely, if M ∈ CPRL satisfies x ∨ y ≈ ( x \ y ) \ y and x ∧ ≈ x ⊕ y ≔ ( x \ \ y and ¬ x ≔ x \
0, then h M , ⊕ , ¬ , i ∈ MV . Again, since MV = ISP ( MV c ), the corresponding variety of pointed residuated lattices is semilinear. ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS V of pointed residuated lattices Hamiltonian if for some k ∈ N > , V | = ( x ∧ e) k · y ≈ y · ( x ∧ e) k . The following proposition collects some useful facts about Hamiltonian varieties ofpointed residuated lattices proved in [12] (see also [6]).P roposition V be aHamiltonian variety of pointed residuated lattices.(a) For any A ∈ V and a , a , b , b ∈ A ,( a , a ) ∈ Cg A ( b , b ) ⇐⇒ ( b ≡ b ) n ≤ a ≡ a for some n ∈ N . (b) V has equationally definable principal congruences if, and only if, for some n ∈ N , V | = ( x ∧ e) n ≈ ( x ∧ e) n + . (c) V has the congruence extension property.We show now that every Hamiltonian variety of pointed residuated lattices has boththe compact intersection property and a guarded deduction theorem, which, combinedwith Theorems 4.5 and 5.7, will allow us to determine which of these varieties have atheory admitting a model completion.L emma V be a Hamiltonian variety of pointed residuated lattices. Then V has the compact intersection property.P roof . Every compact congruence of an algebra A ∈ V is a finite join of principalcongruences of A and hence, by congruence distributivity, the intersection of any twocompact congruences of A is a finite join of intersections of principal congruences of A . It therefore su ffi ces to show that for all b , b , c , c ∈ A ,Cg A ( b , b ) ∩ Cg A ( c , c ) = Cg A (e , ( b ≡ b ) ∨ ( c ≡ c )) . Suppose first that ( a , a ) ∈ Cg A ( b , b ) ∩ Cg A ( c , c ). Then Proposition 6.3.(a) yields( b ≡ b ) n ≤ a ≡ a and ( c ≡ c ) n ≤ a ≡ a for some n , n ∈ N . Let n ≔ max ( n , n ). Then ( b ≡ b ) n ∨ ( c ≡ c ) n ≤ a ≡ a and, using some basicproperties of pointed residuated lattices, also (( b ≡ b ) ∨ ( c ≡ c )) n ≤ a ≡ a . ByProposition 6.3.(a) again, ( a , a ) ∈ Cg A (e , ( b ≡ b ) ∨ ( c ≡ c )). Now suppose that( a , a ) ∈ Cg A (e , ( b ≡ b ) ∨ ( c ≡ c )). It follows easily using some basic properties ofresiduated lattices that ( a , a ) ∈ Cg A ( b , b ) ∩ Cg A ( c , c ). ⊣ L emma V be a Hamiltonian variety of pointed residuated lattices. Then V has a guarded deduction theorem.P roof . We show that the formulas γ ≔ ( x ≡ x ) ≤ z & ( y ≡ y ) ∨ z ≈ e and ϕ ≔ z ≈ esatisfy conditions (i) and (ii) in the definition of a guarded deduction theorem for a finiteset w with z < w , conjunction of equations π ( w ), and terms s ( w ) , s ( w ) , t ( w ) , t ( w ). Generalizing the notion of a Hamiltonian group, an algebraic structure A is usually called Hamiltonian ifevery non-empty subuniverse of A is an equivalence class of some congruence of A . In [6], it is shown thata variety of pointed residuated lattices satisfying x \ e ≈ e / x consists of Hamiltonian algebras in this sense if,and only if, it has the property given in our definition. GEORGE METCALFE AND LUCA REGGIO
For (i), suppose first that V | = ( π & ( t ≈ t )) → ( s ≈ s ). To show that V | = π →∀ z . ( γ → ϕ )( s , s , t , t , z ), consider any A ∈ V and assignment f : w → A such that A , f | = π , and let g : w , z → A be a map extending f such that A , g | = γ ( s , s , t , t , z ).Then A , g | = ( s ≡ s ) ≤ z and A , g | = ( t ≡ t ) ∨ z ≈ e. It follows that also A , g | = z ≤ eand hence to show that A , g | = ϕ ( s , s , t , t , z ), it su ffi ces to prove the following:C laim . A , g | = e ≤ z .P roof of C laim . Let ˜ g : Tm ( w , z ) → A be the unique homomorphism extending g ,and define a ≔ ˜ g ( s ), a ≔ ˜ g ( s ), b ≔ ˜ g ( t ), b ≔ ˜ g ( t ), and c ≔ ˜ g ( z ). We prove firstthat e ≤ ( b ≡ b ) k ∨ c for all k ∈ N > , proceeding by induction on k . The base casefollows from the fact that A , g | = ( t ≡ t ) ∨ z ≈ e. For the inductive step, we obtaine ≤ ( b ≡ b ) k ∨ c by the induction hypothesis ≤ ((( b ≡ b ) ∨ c ) · ( b ≡ b ) k ) ∨ c since e ≤ ( b ≡ b ) ∨ c = (( b ≡ b ) k + ∨ c · ( b ≡ b ) k ) ∨ c by the distributivity of · over ∨≤ ( b ≡ b ) k + ∨ c since b ≡ b ≤ e.Now let q denote the natural quotient map from A onto A / Cg A ( b , b ). By assumption, V | = ( π & ( t ≈ t )) → ( s ≈ s ), so A / Cg A ( b , b ) , q f | = s ≈ s , i.e., ( a , a ) ∈ Cg A ( b , b ). By Proposition 6.3.(a), we obtain ( b ≡ b ) n ≤ a ≡ a for some n ∈ N .But A , g | = ( s ≡ s ) ≤ z , so also ( a ≡ a ) ≤ c and ( b ≡ b ) n ≤ c . Hence e ≤ ( b ≡ b ) n ∨ c = c ; that is, A , g | = e ≤ z . ⊣ Now suppose that V | = π → ∀ z . ( γ → ϕ )( s , s , t , t , z ). To show that V | = ( π & ( t ≈ t )) → ( s ≈ s ), consider any A ∈ V and assignment f : w → A such that A , f | = π & ( t ≈ t ). Since z < w , we can extend f to an assignment g : w , z → A by setting g ( z ) ≔ ˜ f ( s ≡ s ), where ˜ f : Tm ( w ) → A is the unique homomorphism extending f .Clearly, A , g | = π & γ ( s , s , t , t , z ) and hence, by assumption, A , g | = z ≈ e. It followsthat A , f | = ( s ≡ s ) ≈ e, and so A , f | = s ≈ s .For the non-trivial direction of (ii), suppose that V | = ( π & γ ( s , s , t , t , z )) → σ . Toshow that V | = π → σ , consider any equation σ ( w ), algebra A ∈ V , and assignment f : w → A such that A , f | = π . Extend f to an assignment g : w , z → A by setting g ( z ) ≔ e. Then A , g | = π & γ ( s , s , t , t , z ) and hence, by assumption, A , f | = σ . ⊣ Theorems 4.5 and 5.7 then yield the following description of Hamiltonian varietiesof pointed residuated lattices whose theories admit a model completion.T heorem V be a Hamiltonian variety of pointed residuated lattices. Then thetheory of V admits a model completion if, and only if, V is coherent and has equationallydefinable principal congruences, the amalgamation property, and the equational variablerestriction property.E xample ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS roposition V be a Hamiltonian variety of pointed resid-uated lattices that is closed under canonical extensions. If V is coherent, then it hasequationally definable principal congruences.Note that the varieties of lattice-ordered abelian groups and MV-algebras are notclosed under canonical extensions, and the same is true for the classes consisting oftheir linearly ordered members. Indeed, both these varieties are coherent, but do nothave equationally definable principal congruences. Moreover, as mentioned already,the theories of the classes of linearly ordered members of these varieties have a modelcompletion, but this is not the case for the varieties themselves.However, we may use Proposition 6.8 together with the variable projection propertyto establish that the theories of the classes of linearly ordered members of certain otherHamiltonian varieties of pointed residuated lattices do not have a model completion.L emma V be a variety of semilinear pointed residuated lattices. If V c hasthe variable projection property, then V is coherent and V c has the equational variableprojection property.P roof . The claim clearly holds if V c is trivial (contains only trivial algebras), so letus assume that this is not the case. Suppose that V c has the variable projection propertyand consider a finite set x , y and conjunction of equations ϕ ( x , y ). By assumption, thereexists a quantifier-free formula ξ ( x ) such that V c | = ϕ → ξ and for any equation ε ( x ), V c | = ϕ → ε = ⇒ V c | = ξ → ε. We may assume without loss of generality that ξ is a conjunction of formulas of theform π → δ where π is a (possibly empty) conjunction of equations and δ is a non-emptydisjunction of equations. Using some basic properties of pointed residuated lattices, wemay also assume that δ is of the form e ≤ s g · · · g e ≤ s n . But also, using the fact that V c consists of linearly ordered pointed residuated lattices, V c | = (e ≤ s g · · · g e ≤ s n ) ↔ (e ≤ s ∨ · · · ∨ s n ) . Hence we may further assume that ξ is a conjunction of quasiequations. So V = ISP ( V c )has the variable projection property and, by Proposition 2.3, is coherent. But then also V c has the equational variable projection property. ⊣ P roposition V be a Hamiltonian variety of semilinear pointed residuatedlattices that is closed under canonical extensions. If V c has a model completion, then V has equationally definable principal congruences.P roof . Suppose that V c has a model completion. It follows by Proposition 3.8 that V c has the variable projection property and, by the previous lemma, that V is coherent.Hence, by Proposition 6.8, V has equationally definable principal congruences. ⊣ E xample §7. Extending the language. Let L be the language of pointed residuated latticesand let L ⊲ be L extended with an additional binary operation symbol ⊲ . In this section,we show how to associate with any semilinear variety V of pointed residuated lattices, a4 GEORGE METCALFE AND LUCA REGGIO variety V ⊲ in the language L ⊲ that has equationally definable principal congruences andsatisfies the same universal L -sentences as V . We then show that if V admits a certainsyntactic property, the theory of V ⊲ has a model completion. In particular, this is thecase for the varieties of lattice-ordered abelian groups and MV-algebras.Given any variety V of semilinear pointed residuated lattices, let V c ⊲ denote the classof linearly ordered members of V expanded with a binary operation ⊲ defined by x ⊲ y ≔ y if e ≤ x e otherwise.That is, V c ⊲ is the positive universal class consisting of L ⊲ -algebras that satisfy theequational theory of V and the universal sentences ∀ x , y ( x ≤ y g y ≤ x ) and ∀ x , y [(e ≤ x → x ⊲ y ≈ y ) & (e (cid:2) x → x ⊲ y ≈ e)] . Let V ⊲ be the variety generated by V c ⊲ . Since V c ⊲ is a positive universal class, it followsfrom J´onsson’s Lemma [22] that V ⊲ = ISP ( V c ⊲ ). Moreover, we obtain the followingconservative extension result.P roposition V be any variety of semilinear pointed residuated lattices. Thenfor any quantifier-free L -formula χ , V ⊲ | = χ ⇐⇒ V | = χ. P roof . Since V and V ⊲ are both varieties, they satisfy the disjunction property, andit su ffi ces to consider the case where χ is a quasiequation. Suppose first that V ⊲ = χ .Since the L -reduct of any member of V ⊲ belongs to V , also V = χ . Now suppose that V = χ . Since V is semilinear, V = ISP ( V c ), and hence V c = χ . But every member of V c is the L -reduct of a member of V ⊲ , and therefore also V ⊲ = χ . ⊣ For convenience of notation, let us define for any class K of algebras with a pointedresiduated lattice reduct and a finite set of L -terms or L ⊲ -terms Γ ∪ { t } , Γ | = K t : ⇐⇒ K | = & { e ≤ s | s ∈ Γ } → (e ≤ t ) . It follows easily that for any conjunction of equations π and equation u ≈ v , K | = π → u ≈ v ⇐⇒ { s ≡ t | s ≈ t is an equation of π } | = K u ≡ v . Now we use this notation to describe a deduction theorem for V ⊲ , showing that thisvariety has equationally definable principal congruences.P roposition V be a semilinear variety of pointed residuated lattices. Thenfor any finite set x and finite set Γ ∪ { s , t } ⊆ Tm L ⊲ ( x ), Γ ∪ { s } | = V ⊲ t ⇐⇒ Γ | = V ⊲ s ⊲ t . Hence V ⊲ has equationally definable principal congruences.P roof . Using the fact that V ⊲ = ISP ( V c ⊲ ), it su ffi ces to prove that for any finite set x and finite set Γ ∪ { s , t } ⊆ Tm L ⊲ ( x ), Γ ∪ { s } | = V c ⊲ t ⇐⇒ Γ | = V c ⊲ s ⊲ t . That V ⊲ has equationally definable principal congruences follows by Proposition 4.3.Suppose first that Γ ∪ { s } | = V c ⊲ t and consider any A ∈ V c ⊲ and assignment f : x → A satisfying e ≤ ˜ f ( u ) for all u ∈ Γ . If ˜ f ( s ) < e, then e = ˜ f ( s ) ⊲ ˜ f ( t ) = ˜ f ( s ⊲ t );otherwise e ≤ ˜ f ( s ) and, by assumption, e ≤ ˜ f ( t ) = ˜ f ( s ) ⊲ ˜ f ( t ) = ˜ f ( s ⊲ t ). Hence ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS Γ | = V c ⊲ s ⊲ t . Now suppose that Γ | = V c ⊲ s ⊲ t and consider any A ∈ V c ⊲ and assignment f : x → A satisfying e ≤ ˜ f ( s ) and e ≤ ˜ f ( u ) for all u ∈ Γ . Then, by assumption,e ≤ ˜ f ( s ⊲ t ) = ˜ f ( s ) ⊲ ˜ f ( t ) = ˜ f ( t ). Hence Γ ∪ { s } | = V c ⊲ t . ⊣ It will also be useful to have a method for transforming a disjunct in the conclusionof a consequence into a premise. For a finite set x = { x , . . . , x n } and s ∈ Tm L ⊲ ( x ), let ∇ s ≔ s ⊲ (cid:0) (0 ≡ e) ∧ ^ ≤ j ≤ n ( x j ≡ e) (cid:1) . L emma V be a semilinear variety of pointed residuated lattices. Then for anyfinite set x and finite set Γ ∪ { s , t } ⊆ Tm L ⊲ ( x ), Γ | = V ⊲ s ∨ t ⇐⇒ Γ ∪ {∇ s } | = V ⊲ t . P roof . Using the fact that V ⊲ = ISP ( V c ⊲ ), it su ffi ces to prove that for any finite set x = { x , . . . , x n } and finite set Γ ∪ { s , t } ⊆ Tm L ⊲ ( x ), Γ | = V c ⊲ s ∨ t ⇐⇒ Γ ∪ {∇ s } | = V c ⊲ t . Suppose first that Γ | = V c ⊲ s ∨ t and consider any A ∈ V c ⊲ and assignment f : x → A suchthat e ≤ ˜ f ( ∇ s ) and e ≤ ˜ f ( u ) for all u ∈ Γ . Then, by assumption, e ≤ ˜ f ( s ∨ t ). If e ≤ ˜ f ( s ),then e ≤ ˜ f ((0 ≡ e) ∧ V ≤ j ≤ n ( x j ≡ e)), which implies 0 = ˜ f ( x ) = · · · = ˜ f ( x n ) = eand, inductively, ˜ f ( t ) = e. Otherwise, e ≤ ˜ f ( t ). So Γ ∪ {∇ s } | = V c ⊲ t . Now suppose that Γ ∪ {∇ s } | = V c ⊲ t and consider any A ∈ V c ⊲ and assignment f : x → A such that e ≤ ˜ f ( u )for all u ∈ Γ . If e ≤ ˜ f ( s ), then e ≤ ˜ f ( s ∨ t ). Otherwise, ˜ f ( ∇ s ) = e and, by assumption,e ≤ ˜ f ( t ), yielding e ≤ ˜ f ( s ∨ t ). Hence Γ | = V c ⊲ s ∨ t . ⊣ The next lemma provides a method for eliminating occurrences of ⊲ from an L ⊲ -quasiequation while preserving validity in V ⊲ . To avoid multiple case distinctions, weintroduce an extra symbol Λ , fixing s ∨ Λ ≔ s , Λ ∨ s ≔ s , ∇ Λ ≔ e for any L ⊲ -term s .L emma x , s ∈ Tm L ⊲ ( x ), and t ∈ Tm L ⊲ ( x ) ∪ { Λ } , there exista finite set I and s ′ i ∈ Tm L ( x ), t ′ i ∈ Tm L ( x ) ∪ { Λ } for each i ∈ I such that for any u , v ∈ Tm L ⊲ ( x , y ), { u ∧ s } | = V ⊲ t ∨ v ⇐⇒ { u ∧ s ′ i } | = V ⊲ t ′ i ∨ v for all i ∈ I . P roof . Let s ′ [ t ′ ] denote the result of replacing a distinguished occurrence of a vari-able in a term s ′ by a term t ′ . Observe that for s = s ′ [ s ⊲ s ] and any u , v ∈ Tm L ⊲ ( x , y ), { u ∧ s } | = V ⊲ t ∨ v ⇐⇒ { u ∧ s ′ [ s ] ∧ s } | = V ⊲ t ∨ v and { u ∧ s ′ [e] } | = V ⊲ t ∨ s ∨ v . Similarly, for t = t ′ [ t ⊲ t ] and any u , v ∈ Tm L ⊲ ( x , y ), { u ∧ s } | = V ⊲ t ∨ v ⇐⇒ { u ∧ s ∧ t } | = V ⊲ t ′ [ t ] ∨ v and { u ∧ s } | = V ⊲ t ′ (e) ∨ t ∨ v . Iterating these steps yields the required finite set I and s ′ i , t ′ i ∈ Tm L ( x ) for i ∈ I . ⊣ We now provide su ffi cient conditions for a semilinear variety of pointed residuatedlattices V to ensure that the theory of V ⊲ has a model completion.T heorem V be a semilinear variety of pointed residuated lattices such thatfor any finite set x , y and s ∈ Tm L ( x , y ), t ∈ Tm L ( x , y ) ∪ { Λ } , there exist s ′ ∈ Tm L ( x ), t ′ ∈ Tm L ( x ) ∪ { Λ } satisfying for any u , v ∈ Tm L ( x , z ), { u ∧ s } | = V t ∨ v ⇐⇒ { u ∧ s ′ } | = V t ′ ∨ v . Then the theory of V ⊲ has a model completion.6 GEORGE METCALFE AND LUCA REGGIO P roof . It follows from Proposition 7.2 that V ⊲ has equationally definable principalcongruences. Hence, to conclude using Proposition 4.6 that the theory of V ⊲ has amodel completion, it su ffi ces to prove that V ⊲ is coherent and has the amalgamationproperty and the equational variable restriction property.For coherence, it su ffi ces to show that for any finite set x , y = { x , . . . , x n , y } and s ∈ Tm L ⊲ ( x , y ), there exists an s ⋆ ∈ Tm L ⊲ ( x ) such that for any v ∈ Tm L ⊲ ( x ), { s } | = V ⊲ v ⇐⇒ { s ⋆ } | = V ⊲ v . Using Lemma 7.4, there exist a finite set I and s ′ i ∈ Tm L ( x , y ), t ′ i ∈ Tm L ( x , y ) ∪ { Λ } for i ∈ I such that for any v ∈ Tm L ⊲ ( x ), { s } | = V ⊲ v ⇐⇒ { s ′ i } | = V ⊲ t ′ i ∨ v for all i ∈ I . (10)By assumption, there exist s ′′ i ∈ Tm L ( x ), t ′′ i ∈ Tm L ( x ) ∪ { Λ } for each i ∈ I such that forany u ′ , v ′ ∈ Tm L ( x ), { s ′ i ∧ u ′ } | = V t ′ i ∨ v ′ ⇐⇒ { s ′′ i ∧ u ′ } | = V t ′′ i ∨ v ′ . (11)Now consider any v ∈ Tm L ⊲ ( x ). An application of Lemma 7.4 yields a finite set J and u ′ j , v ′ j ∈ Tm L ( x ) for j ∈ J such that for each i ∈ I , { s ′ i } | = V ⊲ t ′ i ∨ v ⇐⇒ { s ′ i ∧ u ′ j } | = V ⊲ t ′ i ∨ v ′ j for all j ∈ J (12) { s ′′ i } | = V ⊲ t ′′ i ∨ v ⇐⇒ { s ′′ i ∧ u ′ j } | = V ⊲ t ′′ i ∨ v ′ j for all j ∈ J . (13)Putting these equivalences together, we obtain { s } | = V ⊲ v ⇐⇒ { s ′ i } | = V ⊲ t ′ i ∨ v for all i ∈ I by (10) ⇐⇒ { s ′ i ∧ u ′ j } | = V ⊲ t ′ i ∨ v ′ j for all i ∈ I , j ∈ J by (12) ⇐⇒ { s ′ i ∧ u ′ j } | = V t ′ i ∨ v ′ j for all i ∈ I , j ∈ J Proposition 7.1 ⇐⇒ { s ′′ i ∧ u ′ j } | = V t ′′ i ∨ v ′ j for all i ∈ I , j ∈ J by (11) ⇐⇒ { s ′′ i ∧ u ′ j } | = V ⊲ t ′′ i ∨ v ′ j for all i ∈ I , j ∈ J Proposition 7.1 ⇐⇒ { s ′′ i } | = V ⊲ t ′′ i ∨ v for all i ∈ I by (13) ⇐⇒ { s ′′ i ∧ ∇ t ′′ i } | = V ⊲ v for all i ∈ I Lemma 7.3 ⇐⇒ { s ′′ i ∧ _ i ∈ I ∇ t ′′ i } | = V ⊲ v . Hence the desired term is s ⋆ ≔ s ′′ i ∧ W i ∈ I ∇ t ′′ i .For the amalgamation property and the equational variable restriction property, itsu ffi ces by Corollary 4.10 to show that for any finite set x , y and t ∈ Tm L ⊲ ( x , y ), either { w } | = V ⊲ t for all w ∈ Tm L ⊲ ( x ) or there exists a t ⋆ ∈ Tm L ⊲ ( x ) such that for any u ∈ Tm L ⊲ ( x , z ), { u } | = V ⊲ t ⇐⇒ { u } | = V ⊲ t ⋆ . Suppose that { w } | = V ⊲ t for some w ∈ Tm L ⊲ ( x ). Using Lemma 7.4 with s = e, there exista finite set I and s ′ i , t ′ i ∈ Tm L ( x , y ) for i ∈ I such that for any u ∈ Tm L ⊲ ( x , z ), { u } | = V ⊲ t ⇐⇒ { u ∧ s ′ i } | = V ⊲ t ′ i for all i ∈ I . (14)Consider now each i ∈ I . By assumption, there exist s ′′ i ∈ Tm L ( x ), t ′′ i ∈ Tm L ( x ) ∪ { Λ } such that for any u ′ , v ′ ∈ Tm L ( x , z ), { s ′ i ∧ u ′ } | = V t ′ i ∨ v ′ ⇐⇒ { s ′′ i ∧ u ′ } | = V t ′′ i ∨ v ′ . (15) ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS { w } | = V ⊲ t , an application of (14) yields { w ∧ s ′ i } | = V ⊲ t ′ i . Defining t ′′′ i ≔ t ′′ i ∨ w , itfollows, using Proposition 7.1 and (15), that for any u ′ , v ′ ∈ Tm L ( x , z ), { s ′ i ∧ u ′ } | = V ⊲ t ′ i ∨ v ′ ⇐⇒ { s ′′ i ∧ u ′ } | = V ⊲ t ′′′ i ∨ v ′ . (16)Now consider any u ∈ Tm L ⊲ ( x , z ). An application of Lemma 7.4 yields a finite set J and u ′ j ∈ Tm L ( x , z ), v ′ j ∈ Tm L ( x , z ) ∪ { Λ } for j ∈ J such that for any i ∈ I , { u ∧ s ′ i } | = V ⊲ t ′ i ⇐⇒ { u ′ j ∧ s ′ i } | = V ⊲ t ′ i ∨ v ′ j for all j ∈ J (17) { u ∧ s ′′ i } | = V ⊲ t ′′′ i ⇐⇒ { u ′ j ∧ s ′′ i } | = V ⊲ t ′′′ i ∨ v ′ j for all j ∈ J . (18)Putting these equivalences together, we obtain for any u ∈ Tm L ⊲ ( x , z ), { u } | = V ⊲ t ⇐⇒ { u ∧ s ′ i } | = V ⊲ t ′ i for all i ∈ I by (14) ⇐⇒ { u ′ j ∧ s ′ i } | = V ⊲ t ′ i ∨ v ′ j for all i ∈ I , j ∈ J by (17) ⇐⇒ { u ′ j ∧ s ′′ i } | = V ⊲ t ′′′ i ∨ v ′ j for all i ∈ I , j ∈ J by (16) ⇐⇒ { u ∧ s ′′ i } | = V ⊲ t ′′′ i for all i ∈ I by (18) ⇐⇒ { u } | = V ⊲ s ′′ i ⊲ t ′′′ i for all i ∈ I Proposition 7.2 ⇐⇒ { u } | = V ⊲ ^ i ∈ I ( s ′′ i ⊲ t ′′′ i ) . Hence the desired term is t ⋆ ≔ V i ∈ I ( s ′′ i ⊲ t ′′′ i ). ⊣ In particular, the conditions of Theorem 7.5 are satisfied when V is the variety LA oflattice-ordered abelian groups.T heorem LA ⊲ has a model completion.P roof . Recall that LA is generated as a quasivariety by the lattice-ordered abeliangroup R = h R , min , max , + , − , i . Hence, by Theorem 7.5, it su ffi ces to prove that forany finite set x , y and s ∈ Tm L ( x , y ), t ∈ Tm L ( x , y ) ∪ { Λ } , there exist s ′ ∈ Tm L ( x ), t ′ ∈ Tm L ( x ) ∪ { Λ } satisfying for any u , v ∈ Tm L ( x , z ), { u ∧ s } | = R t ∨ v ⇐⇒ { u ∧ s ′ } | = R t ′ ∨ v . Note first that if s = s ∨ s and t = t ∧ t and there exist s ′ , s ′ ∈ Tm L ( x ), t ′ , t ′ ∈ Tm L ( x ) ∪ { Λ } satisfying for any u , v ∈ Tm L ( x , z ) and i , j ∈ { , } , { u ∧ s i } | = R t j ∨ v ⇐⇒ { u ∧ s ′ i } | = R t ′ j ∨ v , then s ′ ≔ s ′ ∨ s ′ and t ′ ≔ t ′ ∧ t ′ satisfy the required condition. Let us also note thatif s = s ∧ s and t = t ∨ t , where s , t ∈ Tm L ( x ), and there exist s ′ ∈ Tm L ( x ), t ′ ∈ Tm L ( x ) ∪ { Λ } satisfying for any u , v ∈ Tm L ( x , z ), { u ∧ s } | = R t ∨ v ⇐⇒ { u ∧ s ′ } | = R t ′ ∨ v , then s ′ ≔ s ∧ s ′ and t ′ ≔ t ∨ t ′ satisfy the required condition.Hence using the distributivity properties of R , we may restrict our attention to thecase where s is the meet of at least one of s + ky and s − ky and t is Λ or the join of atleast one of t + ky and t − ky , where k ∈ N > , and s , s , t , t ∈ Tm L ( x ). We construct s ′ ∈ Tm L ( x ), t ′ ∈ Tm L ( x ) ∪ { Λ } as follows. Let s ′ ≔ s + s if s = ( s + ky ) ∧ ( s − ky );otherwise s ′ ≔ e. The term t ′ is Λ or the join of (i) t + t if t = ( t + ky ) ∨ ( t − ky ); (ii) t − s if s + ky occurs in s and t + ky occurs in t ; (iii) t − s if s − ky occurs in s and t − ky occurs in t .8 GEORGE METCALFE AND LUCA REGGIO
We check the case where s = ( s + ky ) ∧ ( s − ky ), t = ( t + ky ) ∨ ( t − ky ), s ′ = s + s ,and t ′ = ( t + t ) ∨ ( t − s ) ∨ ( t − s ), other cases being very similar. Let u , v ∈ Tm L ( x , z )and suppose first that { u ∧ s ′ } | = R t ′ ∨ v . Since { s } | = R s ′ and { s ∧ t ′ } | = R t , it followseasily that { u ∧ s } | = R t ∨ v . Now suppose that { u ∧ s ′ } 6| = R t ′ ∨ v ; that is, some assignment f : x , z → R satisfies 0 ≤ ˜ f ( u ), 0 ≤ ˜ f ( s ) + ˜ f ( s ), ˜ f ( t ) + ˜ f ( t ) <
0, ˜ f ( t ) < ˜ f ( s ),˜ f ( t ) < ˜ f ( s ), and ˜ f ( v ) <
0. We extend f to an assignment f ′ : x , y , z → R by defining f ′ ( y ) ≔ min ( ˜ f ( s ) , − ˜ f ( t )) + max ( − ˜ f ( s ) , ˜ f ( t ))2 k . It then follows that 0 ≤ ˜ f ′ ( u ), 0 ≤ ˜ f ′ ( s ) + ˜ f ′ ( ky ), ˜ f ′ ( ky ) ≤ ˜ f ′ ( s ), ˜ f ′ ( t ) + ˜ f ′ ( ky ) < f ′ ( t ) < ˜ f ′ ( ky ) and ˜ f ′ ( v ) <
0. Hence { u ∧ s } 6| = R t ∨ v . ⊣ The variety MV ⊲ of MV-algebras extended with ⊲ is term-equivalent to the variety MV ∆ of MV ∆ -algebras (see, e.g., [20, 29]), generated by linearly ordered MV-algebrasextended with an additional unary operator ∆ defined by ∆ x ≔ x =
10 otherwise.For any algebra in MV ⊲ , we let ∆ x ≔ ( x ⊲ ⊲
0, and for any algebra in MV ∆ , we let x ⊲ y ≔ ¬ ∆ x ⊕ y . The following theorem may therefore be proved along the same linesas Theorem 7.6, using the fact that MV is generated as a quasivariety by the algebra h [0 , , ⊕ , ¬ , i , where x ⊕ y ≔ min (1 , x + y ) and ¬ x ≔ − x .T heorem MV ∆ has a model completion.A di ff erent proof of the preceding result was announced by X. Caicedo at the 2008conference Residuated Structures: Algebra and Logic . Caicedo also presented a proofat the 2019
Latin American Algebra Colloquium that the theory of the class of so-called“pseudo-complemented” lattice-ordered abelian groups has a model completion, whichseems to be related to our (independently-obtained) Theorem 7.6.
Appendix A. A comparison with Wheeler’s characterization theorem.
In thisappendix, we compare our results from Section 3.3 with the necessary and su ffi cientconditions given by Wheeler in [34] for the existence of model completions for universaltheories with finite presentations. Let us note that Wheeler considers arbitrary first-order languages and, although we restrict ourselves here to algebraic languages, it isnot di ffi cult to see that the results of this appendix can be generalized to any first-orderlanguage by replacing equations with atomic formulas.Let K be a universal class of algebras. Following [34, Section 3], a finite presentation of an algebra A ∈ K is a pair ( a , π ) where a ∈ A x is a finite set of generators for A and π ( x ) is a conjunction of equations such that A | = π ( a ) and for any equation ε ( x ), A | = ε ( a ) ⇐⇒ K | = π → ε. Observe that ( a , π ) is a finite presentation of A if, and only if, for any B ∈ K generated by b ∈ B x such that B | = π ( b ) there exists a surjective homomorphism A ։ B sending a to b . An algebra A ∈ K is said to be finitely presented in K if it admits a finite presentation.R emark A.1. If K is a variety, then an algebra A ∈ K is finitely presented in K if, andonly if, it is finitely presented in the usual sense, i.e., A is isomorphic to the quotient ofa finitely generated K -free algebra with respect to a compact congruence. Moreover, if ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS K is a positive universal class and the variety V ≔ HSP ( K ) generated by K is congruencedistributive, then A ∈ K is finitely presented in K precisely when it is a finitely presentedmember of V . Just observe that V = ISP ( K ), by J´onsson’s Lemma [22].A class of algebras K has finite presentations if for every finite set x and conjunctionof equations π ( x ) that is satisfiable in K , there exist an algebra A ∈ K and a tuple a ∈ A x such that ( a , π ) is a presentation of A . All quasivarieties, and in particular all varieties,have finite presentations (cf. [34, Corollary 1, p. 315]), but this is not the case for alluniversal classes, as shown by the following example.E xample A.2. The positive universal class LA c of ordered abelian groups does nothave finite presentations. To see this, recall that LA c generates the variety LA of lattice-ordered abelian groups, which is congruence distributive (cf. Example 2.7). Hence, byRemark A.1, an ordered abelian group is finitely presented in the sense of the abovedefinition if, and only if, it is finitely presented as a lattice-ordered abelian group inthe usual sense. However, any finitely presented ordered abelian group is simple (see,e.g., [18, Theorem 4.A and Corollary 5.2.3]) and there exist finitely generated orderedabelian groups that are not simple, e.g., the lexicographic product R −→× R generated by { (1 , , (0 , } . It follows there cannot be a 2-generated finitely presented ordered abeliangroup with presentation ⊤ ( x , x ).Let us now recall Wheeler’s conservative congruence extension property. Given anyalgebra A ∈ K , let D + ( A ) be the positive diagram of A , i.e., the set of atomic sentencesin the language extended with names for the elements of A that are satisfied in A . Theclass K has the conservative congruence extension property (for finite presentations) if, whenever B admits a finite presentation (( a , b ) , π ( x , y )), A is the subalgebra of B generated by a , the tuple b does not lie in A , and ρ ( x , y ) is a conjunction of negatedequations such that B | = ρ ( a , b ), there exists a quantifier-free formula χ ( x ) satisfyingTh( K ) ∪ D + ( B ) ⊢ ρ ( a , b ) → χ ( a ) , and for every surjective homomorphism h : A ։ A ′ such that A ′ ∈ K and A ′ | = χ ( h ( a ))there exist a B ′ ∈ K extending A ′ and a surjective homomorphism h ′ : B ։ B ′ whoserestriction to A coincides with h , and such that B ′ | = ρ ( h ( a ) , h ′ ( b )). The following proposition shows that, for universal classes with finite presentations,the conservative congruence extension property is equivalent to a strengthening of theconservative model extension property where the variable y is replaced by a tuple y .P roposition A.3. Let K be a universal class of algebras with finite presentations.Then K has the conservative congruence extension property if, and only if, for anyfinite sets x , y and conjunction of literals ψ ( x , y ), there exists a quantifier-free formula χ ( x ) satisfying(i) K | = ψ → χ (ii) for every A ∈ K generated by a ∈ A x such that A | = χ ( a ) and for any equation ε ( x ), K | = ψ + → ε = ⇒ A | = ε ( a ) , there exist an algebra B ∈ K extending A and b ∈ B y such that B | = ψ ( a , b ). Note that Wheeler does not assume that A ′ ∈ K , but uses this in the proof of his main result. GEORGE METCALFE AND LUCA REGGIO P roof . Suppose that K has the conservative congruence extension property. Fix finitesets x , y and a conjunction of literals ψ ( x , y ). If ψ is not satisfiable in K , we can set χ ≔ ⊥ . Hence, assume that ψ is satisfiable in K . Consider B ∈ K and ( a , b ) ∈ B x , y suchthat (( a , b ) , ψ + ) is a finite presentation of B and let A be the subalgebra of B generatedby a . If there is b i ∈ b such that b i ∈ A , then K | = ψ + → y i ≈ t ( x ) for some term t .Replacing y i by t ( x ) in the formula ψ whenever b i ∈ A , we can assume that no elementof b belongs to A . Further, if B | = σ ( a , b ) for some equation σ of ψ − , then K | = ψ + → σ ,contradicting the fact that ψ is satisfiable in K . Hence B | = ρ ( a , b ), where ρ ( x , y ) is theconjunction of the negated equations ¬ σ for σ ranging over the equations of ψ − (hence, ψ = ψ + & ρ ), and so there exists a quantifier-free formula χ ( x ) satisfying the conditionsfor the conservative congruence extension property.We prove that χ satisfies (i) and (ii). For (i), consider any algebra C ∈ K and tuples c ∈ C x and d ∈ C y such that C | = ψ ( c , d ). We must prove that C | = χ ( c ). Let C ′ be thesubalgebra of C generated by c , d , and note that C ′ | = ψ ( c , d ) because ψ is quantifier-free. Since C ′ | = ψ + ( c , d ), there is a surjective homomorphism B ։ C ′ sending a to c and b to d . Hence, with the obvious interpretation of the new constants, C ′ | = D + ( B ).So Th( K ) ∪ D + ( B ) ⊢ ρ → χ and C ′ | = ρ ( c , d ) entail C ′ | = χ ( c ), and therefore C | = χ ( c ).For (ii), consider any algebra A ′ ∈ K generated by a tuple a ′ ∈ ( A ′ ) x such that A ′ | = χ ( a ′ ) and, for all equations ε ( x ), K | = ψ + → ε entails A ′ | = ε ( a ′ ). Then there existsa (unique) homomorphism h : A → A ′ sending a to a ′ . Just observe that, whenever s ( a ) = t ( a ) for two terms s ( x ) , t ( x ), we have B | = s ( a ) ≈ t ( a ) and hence also K | = ψ + → s ( x ) ≈ t ( x ), which implies A ′ | = s ( a ′ ) ≈ t ( a ′ ). Moreover, h is surjective because a ′ generates A ′ . By the conservative congruence extension property, there are B ′ ∈ K extending A ′ and a surjective homomorphism h ′ : B ։ B ′ that extends h and satisfies B ′ | = ρ ( a ′ , h ′ ( b )). Using the fact that B | = ψ + ( a , b ) entails B ′ | = ψ + ( a ′ , h ′ ( b )), weconclude that B ′ | = ψ ( a ′ , h ′ ( b )), as was to be proved.For the converse direction, let B ∈ K be an algebra admitting a finite presentation(( a , b ) , π ( x , y )), let A be the subalgebra of B generated by a , and assume that b does notlie in A . Further, let ρ ( x , y ) be a conjunction of negated equations such that B | = ρ ( a , b ).Define the conjunction of literals ψ ( x , y ) ≔ π & ρ . Then there exists a quantifier-freeformula χ ( x ) satisfying the properties in (i) and (ii). Condition (i) entails easily thatTh( K ) ∪ D + ( B ) ⊢ ρ ( a , b ) → χ ( a ) . Consider next a surjective homomorphism h : A ։ A ′ with A ′ ∈ K and A ′ | = χ ( h ( a )).Note that h ( a ) generates A ′ and, for any equation ε ( x ), K | = π → ε implies B | = ε ( a )and hence also A | = ε ( a ) and A ′ | = ε ( h ( a )). By (ii), there exist B ′ ∈ K extending A ′ anda tuple b ′ ∈ ( B ′ ) y such that B ′ | = ψ ( h ( a ) , b ′ ). Let B ′′ be the subalgebra of B ′ generatedby h ( a ) , b ′ and note that B ′′ ∈ K because K is a universal class. Since B ′′ | = π ( h ( a ) , b ′ ),there exists a surjective homomorphism h ′ : B ։ B ′′ sending a to h ( a ) and b to b ′ . Inparticular, h ′ extends h . Further, B ′′ extends A ′ and satisfies B ′′ | = ρ ( h ( a ) , b ′ ). Hence K has the conservative congruence extension property. ⊣ We are now in a position to compare Wheeler’s characterization of universal theoriesadmitting a model completion with our results from Section 3. Following Wheeler, wesay that a universal class of algebras with finite presentations K is coherent if, whenever ODEL COMPLETIONS FOR UNIVERSAL CLASSES OF ALGEBRAS B ∈ K is finitely presented in K and A is a finitely generated subalgebra of B , then A isfinitely presented in K .R emark A.4. 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