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.
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References
Andréka, H. andR. Thompson, 1988, ‘A Stone-type representation theorem for algebras of relations of higher ranks’,Transactions of Amer. Math. Soc. 309, 671–682.
Benthem, J. van, 1983,Modal Logic and Classical Logic, Bibliopolis, Naples.
Goldblatt, R. I., 1989, ‘Varieties of complex algebras’,Annals of Pure and Applied Logic 38, 173–241.
Goldblatt, R. I., 1991a, “The McKinsey axiom is not canonical’,The Journal of Symbolic Logic 56, 554–562.
Goldblatt, R. I., 1991b, ‘On closure on canonical embedding algebras’, in H. Andréka, J. D. Monk, and I. Németi (eds.),Algebraic Logic, North-Holland, Amsterdam, 217–229.
Henkin, L., 1970, ‘Extending Boolean operations’,Pacific Journal of Mathematics 32, 723–752.
Henkin, L., J. D. Monk, andA. Tarski, 1971, 1985,Cylindric Algebras. Part 1. Part 2. North-Holland, Amsterdam.
Jónsson, B., 1994, ‘On the canonicity of Sahlqvist identities’, Preprint 94-012, Department of Mathematics, Vanderbilt University.
Jónsson, B. andA. Tarski, 1952, ‘Boolean algebras with operators, Part I’,American Journal of Mathematics 73, 891–939. (Also in [Tarski 1986, Vol. 3].)
Maddux, R., 1982, ‘Some varieties containing relation algebras’,Transactions of Amer. Math. Soc. 272, 501–526.
Németi, I., 1991, ‘Algebraizations of quantifier logics, an introductory overview’,Studia Logica 50, 485–570. (A version extended with proofs, intuitive explanations, new developments, is available as Mathem. Inst. Budapest, Preprint 1994.)
Rijke, M. de andY. Venema, 1991, ‘Sahlqvist's Theorem for Boolean Algebras with Operators’, Technical report ML-91-10, ITLI, University of Amsterdam, September.
Sahlqvist, H., 1975, ‘Completeness and correspondence in the first and second-order semantics for modal logic’, in S. Kanger (ed.),Proceedings of the Third Scandinavian Logic Symposium. Uppsala 1973, North-Holland, Amsterdam, 110–143.
Sambin, G. andV. Vaccaro, 1989, ‘A topological proof of Sahlqvist's theorem’,The Journal of Symbolic Logic 54, 992–999.
Tarski, A., 1986,Collected Papers, Birkhauser Verlag.
Venema, Y., 1991,Many-Dimensional Modal Logic, PhD thesis, Department of Mathematics and Computer Science, University of Amsterdam.
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de Rijke, M., Venema, Y. Sahlqvist's theorem for boolean algebras with operators with an application to cylindric algebras. Stud Logica 54, 61–78 (1995). https://doi.org/10.1007/BF01058532
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DOI: https://doi.org/10.1007/BF01058532