On the Representation Theory for Cylindric Algebras

The main purpose of this paper is to give some new sufficient conditions for the representability of infinite dimensional cylindric algebras. We also discuss certain problems and results in the representation theory reported on by Henkin and Tarski in [5]. In general we adopt the notation of [5]. § 1 contains some additional notation, the statement of a representation theorem of Henkin and Tarski frequently used in this paper, and an embedding theorem which throws some light on that representation result. § 2 is devoted mainly to some simple proofs for known results about the general algebraic theory of representable cylindric algebras. Then in § 3 we turn to representation theory proper. The first result of this section gives a sufficient condition for representability in terms of isomorphic reducts of an algebra (this result was independently obtained by Alfred Tarski). Then follows the definition of a new class of cylindric algebras, diagonal cylindric algebras. The main theorem of this paper is that every diagonal cylindric algebra is representable; this result represents a considerable improvement of some previously known representation theorems. Several interesting corollaries are derived from this result.