The Preservation of Sahlqvist Equations in Completions of Boolean Algebras with Operators
Steven Givant, Yde Venema

Abstract:
The preservation of Sahlqvist equations in completions of Boolean algebras 
with operators
Steven Givant, Yde Venema

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 A are completely additive, he showed that the completion of
A always exists and is unique up to isomorphisms over A. Moreover, strictly 
positive equations are preserved under completions: a strictly positive 
equation that holds in A must hold in the completion of 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 which 
shows that arbitrary Sahlqvist equations need not be preserved.