Sahlqvist's theorem for Boolean algebras with operators with an application to cylindric algebras
Sahlqvist's Theorem for modal logic is generalized to Boolean algebras with operators. A class of identities for Boolean algebra with operators is described, called the Sahlqvist identities. Sahlqvist's Theorem in this general setting states that (1) each Sahlqvist identity has a ``first-order translation'', i.e., a first-order sentence that holds in a relational structure if and only if the identity holds in the complex algebra of that structure, and (2) each Sahlqvist identity holds in a Boolean algebra with operators if and only if it holds in the perfect (or canonical) extension of that algebra. The application to cylindric algebras, mentioned in the title, is a proof that a certain identity, due to Henkin, is equivalent to another identity. The proof proceeds by arguing in the atom structure of the perfect extension of any algebra that satisfies one (or the other) identity.
scientific article; zbMATH DE number 724129
scientific article