Permutations, substitutions and finite axiomatizability

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 $RC{A}_{\hat{I}}\pm$, and Halmos' representable polyadic algebras $RP{A}_{\hat{I}}\pm$ both provide algebraic counterparts to first-order logic. In this paper, we show that the usual finite set of polyadic axioms axiomatize $RP{A}_{\hat{I}}\pm$ over $RD{f}_{\hat{I}}\pm$, the diagonal-free subreducts of elements in $RC{A}_{\hat{I}}\pm$. In short: $RP{A}_{\hat{I}}\pm =P{A}_{\hat{I}}\pm +RD{f}_{\hat{I}}\pm$.