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Step by step – Building representations in algebraic logic

Published online by Cambridge University Press:  12 March 2014

Robin Hirsch and
Ian Hodkinson
Robin Hirsch
Affiliation:
Department of Computer Science, University College, Gower Street, London WC1, UK
Ian Hodkinson
Affiliation:
Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK
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Abstract

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable.

An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras.

Other instances of this approach are looked at, and include the step by step method.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 1 , March 1997, pp. 225 - 279
Copyright
Copyright © Association for Symbolic Logic 1997

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