Injective and projective TT-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 TT, the so-called TT-Boolean algebras. These are the duals of Boolean flows, and by analyzing injective TT-Boolean algebras we are able to characterize projective Boolean flows. Moreover, we characterize the projective TT-Boolean algebras in the case that TT is a group. This characterization shows that the existence of nontrivial projective TT-Boolean algebras depends on the properties of TT.