Relation Formulas for Protoalgebraic Equality Free Quasivarieties; Pałasińska’s Theorem Revisited

We provide a new proof of the following Pałasińska's theorem: Every finitely generated protoalgebraic relation distributive equality free quasivariety is finitely axiomatizable. The main tool we use are Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{Q}}$$\end{document}-relation formulas for a protoalgebraic equality free quasivariety Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{Q}}$$\end{document} . They are the counterparts of the congruence formulas used for describing the generation of congruences in algebras. Having this tool in hand, we prove a finite axiomatization theorem for Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{Q}}$$\end{document} when it has definable principal Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{Q}}$$\end{document}-subrelations. This is a property obtained by carrying over the definability of principal subcongruences, invented by Baker and Wang for varieties, and which holds for finitely generated protoalgebraic relation distributive equality free quasivarieties.


Introduction
In abstract algebraic logic the following theorem of Katarzyna Pa lasińska is remarkable [15]: Every protoalgebraic and filter distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many inference rules and axioms. By reformulating it into the context of equality free quasivarieties we have (see Section 2 for definitions).
The aim of this paper is to provide a new proof of this theorem. For this purpose we apply the technique of definable principal Q-subrelations. This is equality free quasivariety counterpart of the definable principal subcongruences technique invented by Kirby Baker and Ju Wang [2]. They used it for providing a very elegant and short proof of the celebrated Baker's theorem: Every finitely generated congruence distributive variety of algebras is finitely axiomatizable [1]. This technique was also successfully applied by authors of this article for quasivarieties. In [21] we obtained a short proof of Pigozzi's theorem: Every finitely generated relative congruence distributive quasivariety of algebras is finitely axiomatizable [24]. Here we go one step further. To this end we first need to fill a gap in the theory of equality free quasivarieties: the lack of a counterpart of the notion of congruence formula. We do it by introducing the notion of a Q-relation formula without equality for a protoalgebraic equality free quasivariety Q. Let us add that this notion is more subtle than that of a congruence formula. Indeed, it works properly only under additional assumptions summarized in Better Universe Theorem 5.2. Here the key property is the definability of Leibniz equalities in Q by a positive formula.
Let us write few words about Pa lasińska's theorem from the perspective of deductive systems. Note that deductive systems correspond to equality free quasivarieties in a language with one relation symbol which is unary [5]. Models in such a language are called matrices and their relations filters. In this context the assumptions of Pa lasińska's theorem are very natural. Namely, filter distributivity may be guaranteed by the existence of disjunction [13] or by the satisfaction of deduction theorem [9,Corrolary 2.6]. In fact, the latter yields also protoalgebraicity: a generalized form of deduction theorem is equivalent to protoalgebraicity [11].
Substantial effort was made to prove finite axiomatization results for deductive systems before Pa lasińska obtained her theorem. Willem Blok and Don Pigozzi proved that if a deductive system S with finitely many nonaxiomatic inference rules Λ is protoalgebraic, filter distributive, and the class of its finitely irreducible matrices is finitely axiomatizable then S can be presented by Λ and finitely many axioms [3,Theorem 4.1]. The last condition of this theorem holds when S is determined by a finite family of finite matrices, i.e., when the equality free quasivariety corresponding to S is finitely generated. Janusz Czelakowski proved that if the assumptions of protoalgebraicity and of finiteness of Λ are dropped, then S may be presented by finitely many axioms together with, possibly, infinitely many nonaxiomatic inference rules [8,Theorem 5.1]. Moreover, he showed that if S posses a disjunction, which implies filter distributivity, and the class of its finitely irreducible matrices is finitely axiomatizable, then S can be presented by finitely many axioms and nonaxiomatic inference rules [7,Theorem 3.2]. This means that the corresponding equality free quasivariety of matrices is finitely axiomatizable [7,Theorem 3.2].
Pa lasińska's theorem yields a result for ordinary quasivarieties in a language containing operation and relation symbols. Let Q = Mod(Σ) be such a quasivariety. Extend the language of Q by one binary relation symbol x ∼ y, and consider the class Q = Mod( Σ) with axiomatization Σ obtained from Σ by replacing in it all occurrences of t ≈ s by t ∼ s, where t and s are arbitrary terms, and by adding axioms guaranteeing that the interpretations of x ∼ y are strict congruences. Clearly, Q is protoalgebraic (even finitely equivalential) equality free quasivariety and Q is finitely axiomatizable, or finitely generated, iff Q is. Moreover, the relation distributivity of Q translates to the relative congruence distributivity of Q in the sense of [18]. Hence the restriction to algebras in Pigozzi's theorem is not necessary.
There is a common opinion that techniques and ideas from general algebra (or rather from quasivariety theory) may carry over to abstract algebraic logic when deductive systems under consideration are protoalgebraic. However, from the perspective of this paper, we see that what is really necessary is the definability of Leibniz equalities by a positive formula (Theorem 5.2). Definability by a set of positive formulas (i.e., protoalgebraicity cf. Theorem 2.4) is not enough as first-order logic would be left and compactness theorem is lost. Note that in the construction described in the preceding paragraph we got the desired definability for free. In fact, in this case, the proof of Pa lasińska's theorem may be simplified a bit. Indeed, Section 5 is then irrelevant, and some definitions may be simplified (Remarks 3.4 and 6.3).
The paper is organized as follows. In Section 2 we gather the needed information about equality free quasivarieties. In Sections 3, 4, 5 and 6 we develop a general theory needed later: Section 3 is devoted to finitely generated and locally finite equality free quasivarieties. In Section 4 we formulate and prove the analogue of Jónsson's Lemma. When we divide the set of equality free quasi-identities axiomatizing an investigated equality free quasivariety into the set of equality free identities and the rest, then Jónsson's Lemma shows how to reduce the first set to a finite one. In Section 6 we define Q-relation formulas without equality, where Q is an equality free quasivariety. In Section 5 we describe conditions for Q under which this definition makes sense. Sections 7, 8 and 9 are devoted to the proof of Pa lasińska's theorem: In Section 7 we say what it means that an equality free quasivariety Q has definable principal Q-subrelations. In Section 8 we prove a finite axiomatization theorem for such equality free quasivarieties. In Section 9 it is showed that a finitely generated protoalgebraic relation distributive equality free quasivariety Q has definable principal Q-subrelations, thus Pa lasińska's theorem is obtained. Here a brilliant argument due to Baker and Wang is used. Finally Appendix contains information about how to obtain the results when we have more than one relation symbols in the language. Most of the paper is written under the restriction that there is just one such symbol. We do so in order not to obscure the reasoning. It should bring the reader's attention to relevant aspects of the theory, not to notational technicalities.
To finish the introduction let us add that the novelty of this paper lies mainly in introducing the proper notion of Q-relation formula for protoalgebraic equality free quasivarieties. With this tool in hand the results are obtained by translating the arguments from [2] and [21].

Toolbox
Here we collect the facts that we need in the paper. We also fix terminology and notation. The reader may consult the introductory paper [4]. It is focused on models with just one relation and is written from the perspective of deductive systems. However it is not difficult to generalize and translate the results obtained there to our setting of equality free quasivarieties. Moreover, results from [12,14] are particularly important for us. Furthermore, there are books about abstract algebraic logic that may serve here [10,16,26]. Finally, basic knowledge about quasivarieties [18,20] (axiomatization, freeness, generation, subdirect irreducibility) may help reading.
We fix a default first-order language L. We assume that L is finite, i.e., it contains only finitely many operation and relation symbols. We also assume that L does not contain equality symbol ≈. By an equality free formula or a formula without equality we mean a first-order formula in L. We do so because relations are more important than operations in our considerations. Notice that in abstract algebraic logic M is traditionally called a matrix, and R a filter, however we decided to stick with model theoretic terminology.
An equality free quasivariety is a class defined by equality free quasiidentities, i.e., by equality free sentences of the form where n is a natural number and ϕ 0 , . . . , ϕ n−1 , ϕ are atomic formulas. The name "quasi-identity" comes from [18,20], however sentences of this kind are also called strict universal Horn sentences [10]. An equality free variety is a class defined by universally quantified equality free atomic formulas. Such formulas are also called equality free identities.
There is a characterization of equality free quasivarieties by the closure on some class operators. Beside commonly known P product, P U ultraproduct, P SD subdirect product and S submodel class operators we need to consider two more C contraction and E expansion class operators. A homomorphism ) for every tuplē a ∈ M and every relation symbol R ∈ L. (Here and in many places we abuse the notation writingā ∈ M instead of a 0 , . . . , We say that M and N are relatives provided N ∈ EC(M) (or equivalently M ∈ EC(N)). Note that if M and N are relatives, then they satisfy the same equality free sentences. A strict congruence of a model M = (A, R) is a congruence α of the algebra A such that M |= R(ā) ↔ R(b) providedā αb for every relation symbol R and every pair of tuplesā,b of elements from M of the lengths equal the arity of R. We alert that this notion is different than the congruences introduced in [18]. The largest strict congruence of M is called Leibniz equality of M and is denoted by Ω(M). Note that (a, b) ∈ Ω(M) if we cannot distinguish a from b in the following sense: there is no equality free formula ϕ(x,z) and a tuplec ∈ M such that M |= ϕ(a,c) and M |= ¬ϕ(b,c). (See e.g. [17] for the origin and philosophical aspects.) Thus Leibniz equalities are definable by a set of formulas. Note however, that the existence of the automorphism of M switching only two elements does not imply that these elements are indistinguishable in the above sense, and are Leibniz equality congruent, as the graph • • shows. We say that a model M is reduced if Ω(M) is equal to the equality relation on M . We use the notation M * = M/Ω(M), a * = a/Ω(M) for a ∈ M , and C * = {M * | M ∈ C}. Models M and N are relatives iff the reduced models M * and N * are isomorphic.
On an algebra A we order all interpretations of relational part L R of the default language L componentwise: R ⊆ L S if for every R ∈ L R the interpretation of R in R is contained in the interpretation of R in S (i.e., if the identity mapping is a homomorphism from (A, S) to (A, R)). For an equality free quasivariety Q let Rel Q (A) be the set all interpretations R of L R such that (A, R) ∈ Q. Note that Rel Q (A) forms an algebraic lattice with componentwise intersections as meets.
in an equality free quasivariety Q is completely irreducible relative to Q iff R is completely meet irreducible in the lattice Rel Q (M). Let Q CI stand for the class of all completely irreducible models relative to Q. Because every lattice Rel Q (M) is algebraic, every model M ∈ Q may be represented as (A, R i ), where all (A, R i ) ∈ Q CI . From this we obtain the following fact.
Lemma 2.2. Let P and Q be equality free quasivarieties. If P CI ⊆ Q, then P ⊆ Q.
An equality free quasivariety Q is protoalgebraic if for every algebra A and R, S ∈ Rel Q (A) the inclusion R ⊆ S yields Ω(A, R) ⊆ Ω(A, S). In particular, the protoalgebraicity guarantees that complete irreducibility in equality free quasivarieties plays the same role as relative subdirect irreducibility in quasivarieties. The following fact may be treated as an exercise. Optionally, the reader may consult [4, Section 9].  (2) Q * = P SD (Q * CI ).
Formulas of the form (∀z) ϕ(x,z), where ϕ is atomic, are called pseudoatomic. We will need the following nontrivial fact. We encourage the reader to check the brilliant proof. Equality free quasivarieties with Leibniz equalities definable by a (finite) set of equality free atomic formulas are called (finitely) equivalential [6]. For instance, the equality free quasivarietyQ constructed from an ordinary one Q in Introduction is finitely equivalential.
Finally, the free model in an equality free quasivariety Q over a set of variables X is constructed as F Q (X) = (T(X), Rel Q (T(X))), where T(X) is an algebra of terms over X in the algebraic part of the default language. Note that if V is the equality free variety generated by Q, i.e., the class satisfying equality free identities true in Q, then F Q (X) = F V (X) for every X.

Finitely generated equality free quasivarieties
An equality free quasivariety Q is locally finite if all finitely generated submodels of every reduced model from Q are finite. Furthermore, Q is finitely generated if it is generated by a finite family of finite models.   Remark 3.4. In an equivalential equality free quasivariety submodels of reduced models are reduced. Hence for them being locally finite is equivalent to having all reduced finitely generated models finite.
If K CI ∩ V is finitely axiomatizable, then K ∩ V is finitely axiomatizable relative to K.

Note that the conditions (2) and (3) hold when K CI ∩ E = I
Proof. By (1) and then by (2), we have By assumption, C is finitely axiomatizable. Hence, by compactness theorem, there exists a finitely axiomatizable equality free variety W ⊇ V such that K ∩ W ⊆ C. We will show that K ∩ V = K ∩ W with the aid of Lemma 2. This and M ∈ C yield M ∈ K CI ∩ V ⊆ K ∩ V.

Better universe
Lemma 5.1. Let Q be a protoalgebraic equality free quasivariety and C ⊆ Q * . If C is axiomatizable, then there exists an equality free positive formula x ∼ y such that (1) x ∼ y defines Leibniz equalities in E(C); (2) for every M ∈ Q, Ω(M) is contained in the interpretation of x ∼ y in M.
Proof. Let C = Mod(Σ) and x y be a set of equality free pseudo-atomic formulas from Theorem 2.4 defining Leibniz equalities in Q. Then The formula x ∼ y = x f y, which is equality free and positive, defines equalities in C. Thus it defines Leibniz equalities in E(C).
Better Universe Theorem 5.2. Let Q be a protoalgebraic equality free quasivariety with Q * axiomatizable. Then there exists a better universe U : an equality free quasivariety such that (1) Q ⊆ U; (2) U is finitely axiomatizable; (3) U is protoalgebraic; (4) there is a positive equality free formula x ∼ y defining Leibniz equalities in U .
Proof. Let x ∼ y be the formula from Lemma 5.1 when C = Q * . Let χ be a sentence such that M |= χ iff the interpretation of x ∼ y in M is a strict congruence. We have Q |= χ, thus there exists a finitely axiomatizable equality free quasivariety U ⊇ Q such that U |= χ. We will show that for M ∈ U the interpretation of We indicate the case when Better Universe Theorem is applicable.
Proposition 5.3. Let Q be a protoalgebraic equality free quasivariety and Q * CI be axiomatizable. Then Q * is axiomatizable and Better Universe Theorem holds for Q.
Proof. Let x ∼ y be the formula from Lemma 5.1 when C = Q * CI . We claim that Q * is definable relative to Q by σ = (∀x, y)[x ∼ y → x ≈ y]. By point (2) in Lemma 5.1, every model from Q satisfying σ is reduced. Conversely, the positivity of x ∼ y yields preservation of satisfaction of σ under taking subdirect products. Thus, by Propositions 2.3 point (2), Q * |= σ follows from Q * CI |= σ.
Everywhere from now on Q is always assumed to satisfy conditions of Better Universe Theorem, and U , x ∼ y are as there.

Q-Relation formulas
Congruence formulas are a key tool in general algebra. Most standard proofs of finite axiomatization results for (quasi)varieties use them. However their counterparts for protoalgebraic equality free quasivarieties were overlooked. The situation is a bit more complicated here and we need to consider two notions: Q-relation formulas with and without equality. We will introduce them in the case when there is only one relation symbol R in the default language because the general case is a bit cumbersome. We will identify the interpretation R of R in M with the subset of M arity(R) . Then (A, R) |= R(ā) means exactlyā ∈ R. We will present the adjustment to the general case in Appendix.
For tuplesā 0 , . . . ,ā n−1 ∈ M of the lengths arity(R), let Note that relations rel M Q (ā 0 , . . . ,ā n−1 ) are compact elements in the lattice Rel Q (M). Relations of the form rel M Q (ā) will be called principal Q-relations. Observe that a model M ∈ Q is completely irreducible relative to Q iff there exists a tuplec ∈ M such that M |= ¬R(c) and whenever M |= ¬R(ā) for someā ∈ M , thenc ∈ rel M Q (ā). Studying definability of principal Q-relations is one of our main concerns in this paper, as is studying definability of principal congruences in general algebra.
A Formula Γ(ȳ,x) is Q-relation formula with equality if it is (equivalent to) an existential positive formula (or a disjunction of primitive positive formulas), possibly with equality, such that A Q-relation formula without equality is any formula Γ that may be obtained obtained from a Q-relation formula with equality Γ by replacing all occurrences of t ≈ s in Γ by t ∼ s, where t and s are arbitrary terms. Note that Q-relation formulas without equality do not have to be equivalent to existential formulas, but, due to the protoalgebraicity, they are positive. This is an important observation that will be useful later. The simplest Q-relation formulas with(out) equality arex ≈ȳ (x ∼ȳ) and R(ȳ).
Lemma 6.1. Let Γ be a Q-relation formula without equality. Then The following conditions are equivalent.
Proof. It is enough to show thatā ∈ S ∈ Rel Q (M). Becauseȳ ∼x is a Q-relation formula without equality,ā ∈ S. Similarly, the fact that R(ȳ) is a Q-relation formula without equality yields R ⊆ S. In order to see that S ∈ Rel Q (M) consider an equality free quasi-identity true in Q q = (∀z) i R(t i (z)) → R(t(z)) .
We need to verify that (A, S) |= q. So assume that M |= Γ i (t i (d),ā) for some Q-relation formulas without equality Γ i and somed ∈ M . Put Then Γ is equivalent to a Q-relation formula without equality and moreover M |= Γ(t(d),ā). Hence (A, S) |= q.
The proof of the equivalence (1)⇔(2) is analogous. Note however that this equivalence is more general. It holds for arbitrary equality free quasivarieties.
Remark 6.3. Note that when Q is finitely equivalential Q-relation formulas without equality are (equivalent to) existential positive equality free formulas satisfying ( Γ ). Thus then they form a subclass of Q-relation formulas with equality. We say that Q has definable principal subrelations (DPSR in short) if there exists a Q-relation formula without equality Γ defining principal Q-subrelations in Q. Proof. This is so because Then Q has DPSR. The converse is true provided Q is locally finite.

Definable principal Q-subrelations
Proof. We will verify the condition from Lemma 7.1. By the finiteness of the default language it follows that there are only finitely many models of cardinality at most m. Thus, by Proposition 6.2, there is a Q-relation formula with equality Γ such that for every N with |N | m and for all . Letb and N be as in point (1). Then N |= Γ(b,ā) and, because Γ is existential, M |= Γ(b,ā). We analogically verify thatc ∈ rel M Q (b) iff M |= Γ(c,b) forc ∈ M . Conversely, by Lemma 7.1 there exists a Q-relation formula with equality Γ(ȳ,x) = (∃z) γ(x,ȳ,z), where γ is quantifier free, defining principal Q-subrelations in Q * . Take m to be the number from Lemma 3.2 for k = (length ofz) + 2 arity(R). Proof. First notice that every model has a relative of the form (T(X), R) for some set X, this is with algebra of terms as its algebra reduct. Moreover, the assumption gives us F Q (X) = F P (X) =: F. Thus it would be enough to show that Rel P (F) = Rel Q (F). But even less is needed. Because both lattices Rel P (F) and Rel Q (F) are algebraic, we just need to show that they have the same compact elements, i.e., rel F Q (t 0 , . . . ,t n−1 ) = rel F P (t 0 , . . . ,t n−1 )

Finite axiomatization theorem
for allt 0 , . . . ,t n−1 ∈ F , n ∈ N. We will verify this equality by induction on n. For n = 0 the equation clearly holds. So assume that it holds for n. Then Here the first and the last equalities hold by the definition, the second follows from the assumption of the lemma, and the third from the induction assumption. Proof. Let Γ be a Q-relation formula without equality witnessing DPSR for Q. By compactness theorem, there is a finitely axiomatizable equality free quasivariety K such that Q ⊆ K ⊆ U, where U is from Better Universe Theorem and Γ is a K-relation formula without equality. We will prove that P := K ∩ V = Q by verifying the condition from Lemma 8.1. So we want to check that rel M Q (ā) = rel M P (ā) forā ∈ M , M = (A, R) ∈ Q. The inclusion rel M P (ā) ⊆ rel M Q (ā) follows from the containment Q ⊆ P. In order to prove the converse one we construct a sequence (R κ ) κ<ρ , where ρ is an ordinal, of relations on M with the following properties: The first condition yields that (R κ ) κ<ρ is not strictly increasing. Hence, by the last two conditions, we obtain rel M Q (ā) ⊆ rel M P (ā). We start by putting R 0 := R. Let κ = λ + 1 and assume that R λ is already defined. Ifā ∈ R λ , then R λ = rel M Q (ā) sinceā ∈ R λ ∈ Rel Q (M). In this case we simply put R κ := R λ . Otherwise there existsb ∈ R λ such that and we define R κ := rel (A,R λ ) Q (b). Since Γ is a K-, and hence a P-relation formula without equality, by Proposition 6.2 we have R κ ⊆ rel  Proof. Because we implicitly assume that Q satisfies the conditions of Better Universe Theorem, Q CI is finitely axiomatizable iff Q * CI is finitely axiomatizable. (⇒) It follows from Lemma 8.3. (⇐) By Lemma 8.2, there exists an equality free quasivariety K ⊆ U, with a finite axiomatization Σ, such that Q = K ∩ V, where V is an equality free variety generated by Q, and Γ is a K-relation formula without equality.
Let δ(ȳ) be a formula such that for every M ∈ U andb ∈ M We may construct δ as follows. For In other words, E is the subclass of K where Γ defines principal K-subrelations. In particular, Q ⊆ E and the condition (1) from Jónsson's Lemma holds. The class I is the subclass of E satisfying the sentence σ from Lemma 8.3. The satisfaction of the conditions (2) and (3) from Jónsson's Lemma follows from Lemma 8.3. Finally, by the assumption, K CI ∩V = Q CI is finitely axiomatizable. This proves that Q is finitely axiomatizable relative to K. Thus, since K is finitely axiomatizable, Q is finitely axiomatizable.  Theorem 9.2. Let Q be a finitely generated protoalgebraic relation distributive equality free quasivariety. Then Q has DPSR. whereȳ S is a tuple of variables of the length equals arity(S), and σ :=

Pa lasińska's theorem
wherex R andz T are tuples of variables of the lengths equal arity(R) and arity(T ) respectively. In the proof of Theorem 8.4 we need to redefine the sentence axiomatizing the class E relative to K. For