Stone representation theorem for Boolean algebras in the topos M Set

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 M Boo 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.