PP200313: Hodkinson, Ian and Venema, Yde (2003) Canonical varieties with no canonical axiomatisation. [Report]

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Abstract
We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or firstorder 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 noncanonical 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 2colourable. 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 noncanonical sentences.
Item Type:  Report 

Report Nr:  PP200313 
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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