Polyadic Equality Algebras Research Articles

Various notions of represetability for cylindric and polyadic algebras

Abstract For β an ordinal, let PEAβ (SetPEAβ) denote the class of polyadic equality (set) algebras of dimension β. We show that for any infinite ordinal α, if is atomic, then for any n < ω, the n-neat reduct of , in symbols , is a completely representable PEAn (regardless of the representability of ). That is to say, for all non-zero , there is a and a homomorphism such that fa(a) ≠ 0 and for any for which exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under ≡∞,ω. Various notions of representability (such as ‘satisfying the Lyndon conditions’, weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CAns satisfying the Lyndon conditions coincides with the class of atomic algebras in ElScNrnCAω, where El denotes ‘elementary closure’ and Sc is the operation of forming complete subalgebras.

On the definition and the representability of quasi‐polyadic equality algebras

We show that the usual axiom system of quasi polyadic equality algebras is strongly redundant. Then, so called non‐commutative quasi‐polyadic equality algebras are introduced (), in which, among others, the commutativity of cylindrifications is dropped. As is known, quasi‐polyadic equality algebras are not representable in the classical sense, but we prove that algebras in are representable by quasi‐polyadic relativized set algebras, or more exactly by algebras in .

Representations of polyadic-like equality algebras

Assuming the usual finite axiom schema of polyadic equality algebras, axiom (P10) is changed to a stronger version. It is proved that infinite dimensional, polyadic equality algebras satisfying the resulting system of axioms are representable. The foregoing stronger axiom is not given with a first order schema. The latter is to be expected knowing the negative results for the Halmos schema axiomatizability of the representable, infinite dimensional, polyadic equality algebras. Furthermore, Halmos’ well-known classical theorem that “locally finite polyadic equality algebras of infinite dimension α are representable” is generalized for locally-\({\mathfrak{m}}\) polyadic equality algebras, where \({\mathfrak{m}}\) is an arbitrary infinite cardinal and \({\mathfrak{m}}\) < α. Also, a neat embedding theorem is proved for the foregoing classes of polyadic-like equality algebras (a neat embedding theorem does not exists for polyadic equality algebras).

Varieties of Algebras without the Amalgamation Property

Let α be an ordinal and κ be a cardinal, both infinite, such that κ ≤ |α|. For τ ∈αα, let sup(τ) = {i ∈ α: τ(i) ≠ i}. Let G κ = {τ ∈αα: |sup(τ)| < κ}. We consider variants of polyadic equality algebras by taking cylindrifications on Γ ⊆ α, |Γ| < κ and substitutions restricted to G κ. Such algebras are also enriched with generalized diagonal elements. We show that for any variety V containing the class of representable algebas and satisfying a finite schema of equations, V fails to have the amalgamation property. In particular, many varieties of Halmos’ quasi-polyadic equality algebras and Lucas’ extended cylindric algebras (including that of the representable algebras) fail to have the amalgamation property.

Neat embeddings as adjoint situations

Looking at the operation of forming neat \(\alpha \)-reducts as a functor, with \(\alpha \) an infinite ordinal, we investigate when such a functor obtained by truncating \(\omega \) dimensions, has a right adjoint. We show that the neat reduct functor for representable cylindric algebras does not have a right adjoint, while that of polyadic algebras is an equivalence. We relate this categorial result to several amalgamation properties for classes of representable algebras. We show that the variety of cylindric algebras of infinite dimension, endowed with the merry go round identities, fails to have the amalgamation property answering a question of Németi’s. We also study two variants of the so-called cylindric polyadic algebras introduced by Ferenczi (all are reducts of polyadic equality algebras, that are also varieties). We show that one is more cylindric than polyadic, and that the other is more polyadic than cylindric. Our classification is determined by results on neat embeddings and amalgamation expressed from the point of view of category theory, thereby witnessing, and, indeed, further emphasizing, the dichotomy between the cylindric and polyadic paradigms. For example, the first class does not have the unique neat embedding property, fails to have the amalgamation property and the neat reduct functor does not have a right adjoint, while the second class has the unique neat emdedding property, the superamalgamation property and the neat reduct functor is strongly invertible. Other results, like first order definability of the class of neat reducts and the class of completely representable algebras, confirming our classification along these lines are presented.

Bare canonicity of representable cylindric and polyadic algebras

We show that for finite n⩾3, every first-order axiomatisation of the varieties of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of non-canonical formulas. We also show that the class of structures for each of these varieties is non-elementary. The proofs employ algebras derived from random graphs.

Finitary Polyadic Algebras from Cylindric Algebras

It is known that every α-dimensional quasi polyadic equality algebra (QPEAα) can be considered as an α-dimensional cylindric algebra satisfying the merrygo- round properties \((CA^{+}_\alpha, \alpha \geq 4)\). The converse of this proposition fails to be true. It is investigated in the paper how to get algebras in QPEA from algebras in CA. Instead of QPEA the class of the finitary polyadic equality algebras (FPEA) is investigated, this class is definitionally equivalent to QPEA. It is shown, among others, that from every algebra in \(CA^{+}_\alpha\) a β-dimensional algebra can be obtained in QPEAβ where \(\beta < \alpha (\beta \geq 4)\) , moreover the algebra obtained is representable in a sense.

On the equational theory of representable polyadic equality algebras

AbstractAmong others we will prove that the equational theory of ω dimensional representable polyadic equality algebras (RPEAω's) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable (and hence the equational theory of this class is finite schema-axiomatizable. as well). We will also show that the complexity of the equational theory of RPEAω is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildikó Sain and Viktor Gyuris [12], the following methodological conclusions will be drawn: The negative properties of polyadic (equality) algebras can be removed by switching from what we call the “polyadic algebraic paradigm” to the “cylindric algebraic paradigm”.

On the equational theory of representable polyadic equality algebras (extended abstract)

Among others we will see that the equational theory of ! dimensional representable polyadic equality algebras (RP EA!’s) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is nite schema-axiomatizable (and hence the equational theory of this class is nite schemaaxiomatizable, as well). We will also see that the complexity of the equational theory of RP EA! is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildik o Sain and Viktor Gyuris [10], the following methodological conclusions will be drawn: the negative properties of polyadic (equality) algebras can be removed by switching from what we call the \polyadic algebraic paradigm" to the \cylindric algebraic paradigm". 1

Complexity of equations valid in algebras of relations part II: Finite axiomatizations

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known examples of algebras of relations are the varieties RCA n of cylindric algebras of n-ary relations, RPEA n of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCA n has to be very complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. Completely analogous statement holds for the case n ⩾ ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNr nCA n + k . We prove that the complementation-free (i.e. positive) subreducts of RCA n do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCA n . We look at all the possible reducts of RCA n and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEA n and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment L n of first order logic. The reason for this is that RCA n and RPEA n are the natural algebraic counterparts of L n while the varieties SNr nCA n + k are in connection with the proof theory of L n . This paper appears in two parts. The first part (Andreka, 1997) contains the non-finite axiomatizability results. The present second part contains finite axiomatizations of some fragments (reducts) together with a figure summarizing the finite and non-finite axiomatizability results in this area and the problems left open.

Complexity of equations valid in algebras of relations part I: Strong non-finitizability

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce and Schröder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known examples of algebras of relations are the varieties RCA n of cylindric algebras of n-ary relations, RPEA n of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCA n has to be very complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. A completely analogous statement holds for the case n ⩾ ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous non-finitizability properties of the larger varieties SNr nCA n + k . We prove that the complementation-free (i.e. positive) subreducts of RCA n do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCA n . We look at all the possible reducts of RCA n and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEA n and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment L n of first order logic. The reason for this is that RCA n and RPEA n are the natural algebraic counterparts of L n while the varieties SNr nCA n + k are in connection with the proof theory of L n . This paper appears in two parts. This is the first part, it contains the non-finite axiomatizability results. The second part contains finite axiomatizability results together with a figure summarizing the results in this area and the problems left open.

A Modal Logic for Quantification and Substitution

The aim of this paper is to study the n-variable fragment of first order logic from a modal perspective. We define a modal formalism called cylindric mirror modal logic, and show how it is a modal version of first order logic with substitution. In this approach, we can define a semantics for the language which is closely related to algebraic logic, as we find Polyadic Equality Algebras as the modal or complex algebras of our system. The main contribution of the paper is a characterization of the intended ‘mirror cubic’ frames of the formalisms and, a consequence of the special form of this characterization, a completeness theorem for these intended frames. As a consequence, we find complete finite yet unorthodox derivation systems for the equational theory of finite-dimensional representable polyadic equality algebras.

Non-finite-axiomatizability results in algebraic logic

This paper deals with relation, cylindric and polyadic equality algebras. First of all it addresses a problem of B. Jónsson. He asked whether relation set algebras can be expanded by finitely many new operations in a “reasonable” way so that the class of these expansions would possess a finite equational base. The present paper gives a negative answer to this problem: Our main theorem states that whenever Rs+ is a class that consists of expansions of relation set algebras such that each operation of Rs+ is logical in Jónsson's sense, i.e., is the algebraic counterpart of some (derived) connective of first-order logic, then the equational theory of Rs+ has no finite axiom systems. Similar results are stated for the other classes mentioned above. As a corollary to this theorem we can solve a problem of Tarski and Givant [87], Namely, we claim that the valid formulas of certain languages cannot be axiomatized by a finite set of logical axiom schemes. At the same time we give a negative solution for a version of a problem of Henkin and Monk [74] (cf. also Monk [70] and Németi [89]).Throughout we use the terminology, notation and results of Henkin, Monk, Tarski [71] and [85]. We also use results of Maddux [89a].Notation. RA denotes the class of relation algebras, Rs denotes the class of relation set algebras and RRA is the class of representable relation algebras, i.e. the class of subdirect products of relation set algebras. The symbols RA, Rs and RRA abbreviate also the expressions relation algebra, relation set algebra and representable relation algebra, respectively.For any class C of similar algebras EqC is the set of identities that hold in C, while Eq1C is the set of those identities in EqC that contain at most one variable symbol. (We note that Henkin et al. [85] uses the symbol EqC in another sense.)

Singularly Cylindric and Polyadic Equality Algebras

Singularly Cylindric and Polyadic Equality Algebras

Singulary cylindric and polyadic equality algebras

This paper is a contribution to the structure and representation theory of finitedimensional cylindric and polyadic equality algebras. The main result is that every singulary algebra is representable, where an algebra is singular y if it is generated by elements supported by singletons. In particular, all prime algebras are representable (a prime algebra has no proper subalgebra, and every cylindric or polyadic equality algebra has a prime subalgebra). Since every polyadic algebra can be embedded in a polyadic equality algebra the results also apply to them. Every infinite-dimensional singulary algebra is locally finite and hence is known to be representable, so we shall be concerned just with the finite-dimensional case, where previously very little was known about representation. It may be mentioned that for any dimension greater than one there are nonrepresentable cylindric and polyadic equality algebras(2). For some special kinds of singulary algebras we obtain a direct construction of the representation, which brings out the structure of the algebra clearly. The methods used essentially constitute an algebraic version of Behmann's solution of the decision problem for the singulary predicate calculus with equality(3). Since only finitely many variables are available, the algebraization has some novel features. The essential ideas can be seen in the case of prime algebras, for which no advanced results about cylindric or polyadic algebras are needed. In the general case, however, use is made of the fact that a cylindric algebra is representable if every finitely generated subalgebra of it is. In the last section of the paper we discuss the logical counterpart of the methods and results of the algebraic part of the paper.