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Strongly representable atom structures of cylindric algebras

Published online by Cambridge University Press:  12 March 2014

Robin Hirsch  and
Ian Hodkinson
Robin Hirsch
Affiliation:
Department of Computer Science, University College London, London, WC1E 6BT, U.K., E-mail: R.Hirsch@cs.ucl.ac.uk, URL: http://www.cs.ucl.ac.Uk/staff/R.Hirsch/
Ian Hodkinson
Affiliation:
Department of Computing, Imperial College London, London, Sw7 2Az, U.K., E-mail: imh@doc.ic.ac.uk, URL: http://www.doc.ic.ac.uk/~imh
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Abstract

A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n ≥ 3, the class of all strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary.

Our proof is based on the following construction. From an arbitrary undirected, loop-free graph Γ, we construct an n-dimensional atom structure , and prove, for infinite Γ, that is a strongly representable cylindric algebra atom structure if and only if the chromatic number of Γ is infinite. A construction of Erdős shows that there are graphs Γk(k < ω) with infinite chromatic number, but having a non-principal ultraproduct ΠDΓk whose chromatic number is just two. It follows that is strongly representable (each k < ω) but is not.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 811 - 828
Copyright
Copyright © Association for Symbolic Logic 2009

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