Boolean product representations of algebras via binary polynomials

We mimic the construction of guarded algebras and show how to extract a Church algebra out of the binary functions on an arbitrary algebra, containing a Church subalgebra of binary polynomial opera- tions. We put to good use the weak Boolean product representations of these Church algebras to obtain weak Boolean product representations of the original algebras. Although we cannot, in general, say much about the factors in these products, we identify a number of sufficient condi- tions for the stalks to be directly indecomposable. As an application, we prove that every skew Boolean algebra is a weak Boolean product of directly indecomposable skew Boolean algebras.