A New Representation Theory: Representing Cylindric-like Algebras by Relativized Set Algebras

Abstract

Representation theorems. As is known, cylindric algebras are not representable in the classical sense (as a subdirect product of cylindric set algebras), in general. But, the Resek-Thompson theorem states that if the system of cylindric axioms is extended by a new axiom, by the merry-goround property (MGR, for short), then the cylindric-like algebra obtained is representable by a cylindric relativized set algebra. Furthermore, if, in additional, axiom (C4) is weakened (only the commutativity of the single substitutions is assumed) then it is representable by an algebra in Dα, (see [And-Tho,88] and [Fer,07a]). This style of representation theorem is closely connected with the completeness theorems based on the Henkin style semantics in mathematical logic. By an r-representation of a cylindric-like algebra we mean a representation by a cylindric-like relativized set algebra.