Atom structures of cylindric algebras and relation algebras Original Research Article

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 RCAn of representable n-dimensional cylindric algebras is not closed under completions. Further, it follows by an argument of Venema that RCAn 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.