Stone representation theorem for Boolean algebras in the topos MSet

Abstract

The famous Stone representation theorem for Boolean algebras represents them as subalgebras of the powerset Boolean algebras. In this paper, we consider a Stone type representation for Boolean algebras in the functor topos Set^M, for a monoid M. We find an adjunction between the category MBoo of Boolean algebras in Set^M and the dual of the topos Set^M. It is proved that each Boolean algebra in Set^M can be embedded into a power of the two element Boolean algebra if and only if M is a group. It is also seen that in contrast to the classic case, in our class of topoi, the Stone representation theorem for Boolean algebras does not coincide with the (internal) prime ideal theorem.