Atom structures of cylindric algebras and relation algebras

Abstract

For any finite n ⩾ 3 there are two atomic n-dimensional cylindric algebras with the same atom structure, with one representable, the other, not. Hence, the complex algebra of the atom structure of a representable atomic cylindric algebra is not always representable, so that the class RCA n of representable n-dimensional cylindric algebras is not closed under completions. Further, it follows by an argument of Venema that RCA n is not axiomatisable by Sahlqvist equations, and hence nor by equations where negation can only occur in constant terms. Similar results hold for relation algebras.