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

Studia Logica 54 (1):61 - 78 (1995)
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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References found in this work BETA
Robert Goldblatt (1991). The McKinsey Axiom is Not Canonical. Journal of Symbolic Logic 56 (2):554-562.
Robert Goldblatt (1989). Varieties of Complex Algebras. Annals of Pure and Applied Logic 44 (3):173-242.
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Leon Henkin (1971). Cylindric Algebras. Amsterdam,North-Holland Pub. Co..
Yde Venema (1995). Cylindric Modal Logic. Journal of Symbolic Logic 60 (2):591-623.
Yde Venema (2003). Atomless Varieties. Journal of Symbolic Logic 68 (2):607-614.
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