Injective and projective T-Boolean algebras

We introduce and analyze Boolean algebras acted on continuously (with respect to the discrete topology on the Boolean algebra) by a topological monoid T, the so-called T-Boolean algebras. These are the duals of Boolean flows, and by analyzing injective T-Boolean algebras we are able to characterize projective Boolean flows. Moreover, we characterize the projective T-Boolean algebras in the case that T is a group. This characterization shows that the existence of nontrivial projective T-Boolean algebras depends on the properties of T.