Polyadic Algebras

Polyadic algebras were introduced and intensively studied by Halmos, after having studied cylindric algebras in Tarski’s seminar in Berkeley; we refer to Section 5.4 of [Hen-Mon-Tar,85], see also [Hal,62]. This class of algebras can be regarded as an alternative approach to algebraize first order logic. After a thorough reformulation of Henkin, Monk, and Tarski, polyadic algebras also can be regarded as certain generalizations of cylindric algebras. On one hand, polyadic algebras have nice representation properties, on the other, their languages are rather large (in the ω-dimensional case the cardinality of their set of operations is continuum), which makes their equational theory recursively undecidable for trivial reasons. This is undesirable from metalogical point of view, hence, during the last decades, certain countable (even finite) reducts of polyadic algebras have also been intensively studied. The goal of this research direction is to find a countable reduct of polyadic algebras which has nice representation properties, and, at the same time, their equational theory is recursively enumerable.