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Studia Logica

, Volume 54, Issue 1, pp 61–78 | Cite as

Sahlqvist's theorem for boolean algebras with operators with an application to cylindric algebras

  • Maarten de Rijke
  • Yde Venema
Article

Abstract

For an arbitrary similarity type of Boolean Algebras with Operators we define a class ofSahlqvist 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 arecanonical, 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 calledHenkin's equation is equivalent to a simpler identity that uses only one variable. We give a conceptually simple proof by showing that the first-order correspondents of these two equations are equivalent over the class of cylindric atom structures.

Keywords

Atom Structure Variety Versus Computational Linguistic Canonical Structure Elementary Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Maarten de Rijke
    • 1
  • Yde Venema
    • 2
  1. 1.CWIAmsterdamThe Netherlands
  2. 2.Dept of Math. and Computer Sc.Free UniversityAmsterdamThe Netherlands

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