Permutations, substitutions and finite axiomatizability

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