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Sahlqvist's Theorem for Boolean Algebras with Operators with an Application to Cylindric Algebras

Maarten de Rijke and Yde Venema
Studia Logica: An International Journal for Symbolic Logic
Vol. 54, No. 1 (Jan., 1995), pp. 61-78
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015763
Page Count: 18
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Sahlqvist's Theorem for Boolean Algebras with Operators with an Application to Cylindric Algebras
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Abstract

For an arbitrary similarity type of Boolean Algebras with Operators we define a class of Sahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities are canonical, that is, their validity is preserved under taking canonical embedding algebras. Taken together, these properties imply that results about a Sahlqvist variety V van be obtained by reasoning in the elementary class of canonical structures of algebras in V. We give an example of this strategy in the variety of Cylindric Algebras: we show that an important identity called Henkin's equation is equivalent to a simpler identity that uses only one variable. We give a conceptually simple proof by showing that the firstorder correspondents of these two equations are equivalent over the class of cylindric atom structures.

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