Summary: 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 Erdős, 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.