Algebraic products of tensor products

Abstract

Given a discrete semigroup $(S,\cdot )$, there is a natural operation on the Stone–Čech 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.