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June 2004 Erdős graphs resolve Fine’s canonicity problem
Robert Goldblatt, Ian Hodkinson, Yde Venema
Bull. Symbolic Logic 10(2): 186-208 (June 2004). DOI: 10.2178/bsl/1082986262

Abstract

We show that there exist 20 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of Thomason). The constructions use the result of Erdős that there are finite graphs with arbitrarily large chromatic number and girth.

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Robert Goldblatt. Ian Hodkinson. Yde Venema. "Erdős graphs resolve Fine’s canonicity problem." Bull. Symbolic Logic 10 (2) 186 - 208, June 2004. https://doi.org/10.2178/bsl/1082986262

Information

Published: June 2004
First available in Project Euclid: 26 April 2004

zbMATH: 1060.03038
MathSciNet: MR2062417
Digital Object Identifier: 10.2178/bsl/1082986262

Keywords: Boolean algebras with operators , canonical extension , elementary class , modal logic , Random graphs , variety

Rights: Copyright © 2004 Association for Symbolic Logic

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Vol.10 • No. 2 • June 2004
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