Sahlqvist's theorem for Boolean algebras with operators with an application to cylindric algebras
type: Item
instance of: scholarly article
title: Sahlqvist's theorem for Boolean algebras with operators with an application to cylindric algebras
zbMATH Open document ID: 0823.03037
DOI: 10.1007/BF01058532
author: Maarten de Rijke
author: Yde Venema
published in: Studia Logica
publication date: 1995-02-22T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 03G15
Mathematics Subject Classification ID: 06E25
zbMATH DE Number: 724129
zbMATH Keywords: Sahlqvist's Theorem
zbMATH Keywords: Boolean algebras with operators
zbMATH Keywords: Sahlqvist identities
zbMATH Keywords: cylindric algebras
reviewed by: Roger Maddux
MaRDI profile type: Publication
cites work: A Stone Type Representation Theorem for Algebras of Relations of Higher Rank
cites work: Determiners and logic
cites work: Varieties of complex algebras
cites work: The McKinsey axiom is not canonical
cites work: Q4011717
cites work: Extending Boolean operations
cites work: Q5616133
cites work: Boolean Algebras with Operators. Part I
cites work: Some Varieties Containing Relation Algebras
cites work: Algebraization of quantifier logics, an introductory overview
cites work: A new proof of Sahlqvist's theorem on modal definability and completeness
cites work: A Modal Logic for Chopping Intervals
full work available at URL: https://doi.org/10.1007/bf01058532
OpenAlex ID: W2033136385
description: scientific article; zbMATH DE number 724129