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Abstract:

Abstract. 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 noncanonical sentences. Using probabilistic methods of Erdős, we construct an infinite sequence G0, G1,... of finite graphs with arbitrarily large chromatic number, such that each Gn is a bounded morphic image of Gn+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 Gn 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. 1.

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