PP-2003-13: Canonical varieties with no canonical axiomatisation

PP-2003-13: Hodkinson, Ian and Venema, Yde (2003) Canonical varieties with no canonical axiomatisation. [Report]

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We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety 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,... 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 V (RRA), while its canonical extension satisfies only a bounded number of them. It now follows by compactness that V (RRA) has no canonical axiomatisation. A variant of this argument shows that there is no axiomatisation using finitely many non-canonical sentences.

Item Type: Report
Report Nr: PP-2003-13
Series Name: Prepublication (PP) Series
Year: 2003
Uncontrolled Keywords: Canonical axiomatisation, canonical equation, canonical modal logic, canonical variety, game, inverse system, random graph, relation algebra
Subjects: Logic
Date Deposited: 12 Oct 2016 14:36
Last Modified: 12 Oct 2016 14:36
URI: https://eprints.illc.uva.nl/id/eprint/97

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