Exploring Canonical Axiomatisations of Representable Cylindric Algebras

be axiomatised by canonical rst-order formulas. So, although RCA_n is known to be
canonical, which means that it is closed under canonical extensions, there is no axiomatisation
where all the formulas are preserved by canonical extensions. In fact, we show
that every axiomatisation contains an innite number of non-canonical formulas.
The proof employs algebras derived from random graphs to construct a cylindric algebra
that satises any number of axioms we want, while its canonical extension only satises
a bounded number. We achieve this by relating the chromatic number of a graph to the
number of RCA_n axioms satised by a cylindric algebra constructed from it.
Finally, we outline a strategy to further generalise the proof to extend the result to
variations of cylindric algebras, such as diagonal-free algebras.