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 D_α, (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.