The preservation of Sahlqvist equations in completions of Boolean algebras with operators

Abstract. Monk [1970] extended the notion of the completion of a Boolean algebra to Boolean algebras with operators. Under the assumption that the operators of such an algebra $ \cal A $ are completely additive, he showed that the completion of $ \cal A $ always exists and is unique up to isomorphisms over $ \cal A $. Moreover, strictly positive equations are preserved under completions a strictly positive equation that holds in $ \cal A $ must hold in the completion of $ \cal A $. In this paper we extend Monk’s preservation theorem by proving that certain kinds of Sahlqvist equations (as well as some other types of equations and implications) are preserved under completions. An example is given that shows that arbitrary Sahlqvist equations need not be preserved.