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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Canonical varieties with no canonical axiomatisation


Authors: Ian Hodkinson and Yde Venema
Journal: Trans. Amer. Math. Soc. 357 (2005), 4579-4605
MSC (2000): Primary 03G15, 08B99; Secondary 03B45, 03C05, 05C15, 05C38, 05C80, 06E15, 91A43
Published electronically: December 16, 2004
MathSciNet review: 2156722
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Abstract: We give a simple example of a variety $\mathbf{V}$ of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety $\mathbf{RRA}$ of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many non-canonical sentences.

Using probabilistic methods of Erdos, we construct an infinite sequence $G_0,G_1,\ldots$ of finite graphs with arbitrarily large chromatic number, such that each $G_n$ is a bounded morphic image of $G_{n+1}$ and has no odd cycles of length at most $n$. The inverse limit of the sequence is a graph with no odd cycles, and hence is 2-colourable. It follows that a modal algebra (respectively, a relation algebra) obtained from the $G_n$ satisfies arbitrarily many axioms from a certain axiomatisation of $\mathbf{V}\ (\mathbf{RRA})$, while its canonical extension satisfies only a bounded number of them. First-order compactness will now establish that $\mathbf{V}\ (\mathbf{RRA})$ has no canonical axiomatisation. A variant of this argument shows that all axiomatisations of these classes have infinitely many non-canonical sentences.


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Additional Information

Ian Hodkinson
Affiliation: Department of Computing, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
Email: imh@doc.ic.ac.uk

Yde Venema
Affiliation: Institute for Logic, Language and Computation, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Email: yde@science.uva.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03743-2
PII: S 0002-9947(04)03743-2
Keywords: Canonical axiomatisation, canonical equation, canonical modal logic, canonical variety, canonicity, chromatic number, game, inverse system, random graph, relation algebra
Received by editor(s): May 23, 2003
Received by editor(s) in revised form: December 19, 2003
Published electronically: December 16, 2004
Additional Notes: The first author’s research was partially supported by grant GR/S19905/01 from the UK EPSRC. He gratefully thanks the members of ILLC of the University of Amsterdam for their warm hospitality during his visit in autumn 2002. The authors thank Robin Hirsch, Agi Kurucz, and Michael Zakharyaschev for valuable conversations, and the referee for helpful comments.
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.