Canonical varieties with no canonical axiomatisation

Abstract

We give a simple example of a varietyV\mathbf {V}of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the varietyRRA\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 sequenceG0,G1,…G_0,G_1,\ldotsof finite graphs with arbitrarily large chromatic number, such that eachGnG_nis a bounded morphic image ofGn+1G_{n+1}and has no odd cycles of length at mostnn. 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 theGnG_nsatisfies arbitrarily many axioms from a certain axiomatisation ofV (RRA)\mathbf {V}\ (\mathbf {RRA}), while its canonical extension satisfies only a bounded number of them. First-order compactness will now establish thatV (RRA)\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.