A polyadic algebra of infinite dimension is completely representable if and only if it is atomic and completely additive

TL;DR

This paper characterizes when infinite-dimensional polyadic algebras are completely representable, showing they are exactly the atomic and completely additive ones, and discusses axiomatization and related algebraic classes.

Contribution

It proves a characterization of complete representability for infinite-dimensional polyadic algebras and provides an infinite first-order axiomatization for this class.

Findings

Complete representability is equivalent to atomicity and complete additivity.

An infinite, uncountable schema axiomatizes the class of completely representable polyadic algebras.

Similar results hold for non-commutative reducts, with some conditions relaxed.

Abstract

We prove the result in the title. We infer, that unlike cylindric algebras, there is a first order axiomatization of the class of completely representable polyadic algebras of infinite dimension, though the one we obtain is infinite; in fact uncountable, but shares a single schema, stipulating that the (uncountably many)substitution operators are completely additive. Similar results are obtained for non commutative reducts of polyadic equality algebras of infinite dimensions, where we can drop complete additivity. However, it remains unknown to us whether there are atomic polyadic algebras of infinite dimension that are not completely additive; but we strongly conjecture that there are.