Atom structures and Sahlqvist equations

This paper addresses the question for which varieties of boolean algebras with operators membership of an atomic algebra \( \frak U \) is determined by its atom structure \( \frak {Ut \,\, U} \). We prove a positive answer for conjugated Sahlqvist varieties; we also show that the conjugation condition is necessary. As a corollary to the positive result and a recent result by I. Hodkinson, we prove that the variety RRA of representable relation algebras, although canonical, cannot be axiomatized by Sahlqvist equations.