Canonical varieties with no canonical axiomatisation |
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Authors: | Ian Hodkinson Yde Venema |
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Affiliation: | Department of Computing, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom ; Institute for Logic, Language and Computation, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands |
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Abstract: | We give a simple example of a variety of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety 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 of finite graphs with arbitrarily large chromatic number, such that each is a bounded morphic image of and has no odd cycles of length at most . 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 satisfies arbitrarily many axioms from a certain axiomatisation of , while its canonical extension satisfies only a bounded number of them. First-order compactness will now establish that 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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Keywords: | Canonical axiomatisation canonical equation canonical modal logic canonical variety canonicity chromatic number game inverse system random graph relation algebra |
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