Algebraic products of tensor products

Given a discrete semigroup $$(S,\cdot )$$ , there is a natural operation on the Stone–Cech compactification $$\beta S$$ of S which extends the operation of S and makes $$(\beta S,\cdot )$$ a compact right topological semigroup with S contained in its topological center. If S and T are discrete semigroups, $$p\in \beta S$$ , and $$q\in \beta T$$ , then the tensor product $$p\otimes q$$ is a member of $$\beta (S\times T)$$ . It is known that tensor products are both algebraically and topologically rare in $$\beta (S\times T)$$ . We investigate when the algebraic product of two tensor products is again a tensor product. We get a simple characterization for a large class of semigroups. The characterization is in terms of a notion of cancellation. We investigate where that notion sits among standard cancellation notions.