Permutations, substitutions and finite axiomatizability

TL;DR

This paper demonstrates that the class of representable polyadic algebras can be finitely axiomatized over diagonal-free reducts of representable cylindric algebras, clarifying their algebraic relationship.

Contribution

It establishes that $RPA_{\alpha}$ is finitely axiomatized over $RDf_{\alpha}$, showing $RPA_{\alpha} = PA_{\alpha} + RDf_{\alpha}$, thus clarifying their algebraic connection.

Findings

Finite axiomatization of $RPA_{\alpha}$ over $RDf_{\alpha}$

Equivalence $RPA_{\alpha} = PA_{\alpha} + RDf_{\alpha}$

Enhanced understanding of algebraic structures for logical systems

Abstract

Algebras of relations form an algebraic framework for the study of logical systems, extending the correspondence between Boolean algebras and propositional logic. Tarski's representable cylindric algebras $RCA_{\alpha}$, and Halmos' representable polyadic algebras $RPA_{\alpha}$ both provide algebraic counterparts to first-order logic. In this paper, we show that the usual finite set of polyadic axioms axiomatize $RPA_{\alpha}$ over $RDf_{\alpha}$, the diagonal-free subreducts of elements in $RCA_{\alpha}$. In short: $RPA_{\alpha} = PA_{\alpha} + RDf_{\alpha}$.