It is known that every α-dimensional quasi polyadic equality algebra (QPEA α ) can be considered as an α-dimensional cylindric algebra satisfying the merrygo- round properties . 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 a β-dimensional algebra can be obtained in QPEA β where , moreover the algebra obtained is representable in a sense. (shrink)

It is shown that a locally finite polyadic algebra on an infinite set V of variables is a Boolean-algebra object, endowed with some internal supremum morphism, in the category of locally finite transformation sets on V. Then, this new categorical definition of polyadic algebras is used to simplify the theory of these algebras. Two examples are given: the construction of dilatations and the definition of terms and constants.

SC, CA, QA and QEA denote the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasi-polyadic algebras and quasi-polyadic equality algebras, respectively. Let ω ≤ α < β and let K ∈ {SC,CA,QA,QEA}. We show that the class of α -dimensional neat reducts of algebras in Kβ is not elementary. This solves a problem in [3]. Also our result generalizes results proved in [2] and [3].

We show that for finite n⩾3n⩾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.

We consider countable so‐called rich subsemigroups of ; each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, if and only if is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in (...) class='Hi'>polyadic equality set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented. (shrink)

This paper is a survey of recent results concerning connections between relation algebras , cylindric algebras and polyadic equality algebras . We describe exactly which subsets of the standard axioms for RA are needed for axiomatizing RA over the RA-reducts of CA3's, and we do the same for the class SA of semi-associative relation algebras. We also characterize the class of RA-reducts of PEA3's. We investigate the interconnections between the RA-axioms within CA3 in more detail, (...) and show that only four implications hold between them . In the other direction, we introduce a natural CA-theoretic equation MGR+, generalization of the well-known Merrry-Go-Round equation MGR of CA-theory. We show that MGR+ is equivalent to the RA-reduct being an SA, and that MGR+ implies that the RA-reduct determines the algebra itself, while MGR is not sufficient for either of these to hold. Then we investigate how different CA's a single RRA can 'generate' in the general case. We solve the first part of Problem 11 from the 'Problem Session Paper' of [2].While proving some of the statements, for others we give only outline of proof. The paper contains several open problems. A full version of this paper is under preparation.Keywords: relation algebras, cylindric algebras, polyadic algebras, algebraic logic, arrow logic, proof theory, finite variable fragments, provability with 3 variables, non-finitizability, twisting, non-standard models, neat reducts, representability. (shrink)

SC, CA, QA and QEA denote the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasi-polyadic algebras and quasi-polyadic equality algebras, respectively. Let ω ≤ α < β and let K ∈ {SC,CA,QA,QEA}. We show that the class of α -dimensional neat reducts of algebras in Kβ is not elementary. This solves a problem in [3]. Also our result generalizes results proved in [2] and [3].

Following research initiated by Tarski, Craig and Nemeti, and futher pursued by Sain and others, we show that for certain subsets G of $^\omega \omega $ , atomic countable G poiyadic algebras are completely representable. G polyadic algebras are obtained by restricting the similarity type and axiomatization of ω-dimensional polyadic algebras to finite quantifiers and substitutions in G. This contrasts the cases of cylindric and relation algebras.

In this note we give an interpretation of cylindric algebras as algebras of sentences of first order logic. We show that the isomorphism types of such algebras of sentences coincide with the class of neat reducts of cylindric algebras. Also we show how this interpretation sheds light on some recent results. This is done by likening Henkin's Neat Embedding Theorem to his celebrated completeness proof.

In this paper we prove polyadic counterparts of the Hájek, Paris and Shepherdson's conservative extension theorems of Łukasiewicz predicate logic to rational Pavelka predicate logic. We also discuss the algebraic correspondents of the provability and truth degree for polyadic MV-algebras and prove a representation theorem similar to the one for polyadic Pavelka algebras.

In 2015, A.V. Figallo and G. Pelaitay introduced tense n×m-valued Łukasiewicz–Moisil algebras, as a common generalization of tense Boolean algebras and tense n-valued Łukasiewicz–Moisil algebras. Here we initiate an investigation into the class tpLMn×m of tense polyadic n × m-valued Łukasiewicz–Moisil algebras. These algebras constitute a generalization of tense polyadic Boolean algebras introduced by Georgescu in 1979, as well as the tense polyadic n-valued Łukasiewicz–Moisil algebras studied by Chiriţă in 2012. (...) Our main result is a representation theorem for tense polyadic n × m-valued Łukasiewicz–Moisil algebras. (shrink)

In this note we give an interpretation of cylindric algebras as algebras of sentences of first order logic. We show that the isomorphism types of such algebras of sentences coincide with the class of neat reducts of cylindric algebras. Also we show how this interpretation sheds light on some recent results. This is done by likening Henkin's Neat Embedding Theorem to his celebrated completeness proof.

In this paper we introduced various classes of weakly associative relation algebras with polyadic composition operations. Among them is the class RWA of representable weakly associative relation algebras with polyadic composition operations. Algebras of this class are relativized representable relation algebras augmented with an infinite set of operations of increasing arity which are generalizations of the binary relative composition. We show that RWA is a canonical variety whose equational theory is decidable.

This paper is a contribution to the algebraic logic of probabilistic models of Łukasiewicz predicate logic. We study the MV-states defined on polyadic MV-algebras and prove an algebraic many-valued version of Gaifman’s completeness theorem.

Among others we will prove that the equational theory of ω dimensional representable polyadic equality algebras (RPEA ω '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 RPEA ω is also extremely high (...) in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildiko 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". (shrink)

Among others we will prove that the equational theory of $\omega$ dimensional representable polyadic equality algebras 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. We will also show that the complexity of the equational theory of RPEA$_\omega$ is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildiko (...) Sain and Viktor Gyuris [12], the following methodological conclusions will be drawn: The negative properties of polyadic algebras can be removed by switching from what we call the "polyadic algebraic paradigm" to the "cylindric algebraic paradigm". (shrink)

Among others we will see that the equational theory of ω dimensional representable polyadic equality algebras. We will also see 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 Ildiko Sain and Viktor Gyuris [10], the following methodological conclusions will be drawn: the negative properties of polyadic algebras can be removed by switching from what we call (...) the `polyadic algebraic paradigm' to the `cylindric algebraic paradigm'. (shrink)

The present note is a reworking and streamlining of Daigneault and Monk's Representation Theory for Polyadic Algebras. MSC: 03G15.

We prove several interpolation theorems for many-valued infinitary logic with quantifiers by studying expansions of MV-algebras in the spirit of polyadic and cylindric algebras. We prove for various reducts of polyadic MV-algebras of infinite dimensions that if is the free algebra in the given signature,, is in the subalgebra of generated by, is in the subalgebra of generated by and, then there exists an interpolant in the subalgebra generated by and such that. We call this (...) a varying interpolation property because the integer depends on the inequality. We also address cases where this interpolation property fails, but other weaker ones hold. One such interpolation theorem says that though an interpolant may not be found as above, an interpolant can always be found if finitely many universal quantifiers are applied to making it smaller and the same number of existential quantifiers are applied to making it bigger. This number of quantifiers also varies; it depends on the inequality. Se.. (shrink)

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.