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


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.


Atom Structure Variety Versus Computational Linguistic Canonical Structure Elementary Class 
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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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