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Relation algebra reducts of cylindric algebras and complete representations

Published online by Cambridge University Press:  12 March 2014

Robin Hirsch
Robin Hirsch*
Affiliation:
Department of Computer Science, University College, Gower Street, London WC1E 6BT, United Kingdom, E-mail: R.Hirsch@cs.ucl.ac.uk, URL: http://www.cs.ucl.ac.Uk/staff/R.Hirsch
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Abstract

We show, for any ordinal γ ≥ 3, that the class ℜaCAγ is pseudo-elementary and has a recursively enumerable elementary theory. ScK denotes the class of strong subalgebras of members of the class K. We devise games, Fn (3 ≤ n ≤ ω), G, H, and show, for an atomic relation algebra with countably many atoms, that

for 3 ≤ n < ω. We use these games to show, for γ > 5 and any class K of relation algebras satisfying

that K is not closed under subalgebras and is not elementary. For infinite γ, the inclusion ℜaCAγ ⊂ ScℜaCAγ is strict.

For infinite γ and for a countable relation algebra we show that has a complete representation if and only if is atomic and ∃ has a winning strategy in F (At()) if and only if is atomic and ∈ ScℜaCAγ.

Keywords

relation algebracylindric algebracomplete representation
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 2 , June 2007 , pp. 673 - 703
Copyright
Copyright © Association for Symbolic Logic 2007

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