Completion of Boolean algebras in MSet

Two important algebraic structures in many branches of mathematics as well as in computer science are M-sets (sets with an action of a monoid M on them) and Boolean algebras. Of particular significance are complete Boolean algebras. And in the absence of the desired completeness one often considers extensions which remedy this lack, preferably in a “universal” way as a normal completion. Combining these two structures one gets M-Boolean algebras (Boolean algebras with an action of M on them, which are a special case of Boolean algebras with operators).The aim of this paper is to study the general notion of an internally complete poset in a topos, in the sense of Johnstone, and use it to give a minimal normal completion for an M-Boolean algebra.