Journal of Symbolic Logic

Step by Step-Building Representations in Algebraic Logic

Robin Hirsch and Ian Hodkinson

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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. Finite relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is $\omega$-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.

Article information

Source
J. Symbolic Logic, Volume 62, Issue 1 (1997), 225-279.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745193

Mathematical Reviews number (MathSciNet)
MR1450522

Zentralblatt MATH identifier
0879.03018

JSTOR
links.jstor.org

Citation

Hirsch, Robin; Hodkinson, Ian. Step by Step-Building Representations in Algebraic Logic. J. Symbolic Logic 62 (1997), no. 1, 225--279. https://projecteuclid.org/euclid.jsl/1183745193


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