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.