Product expansions Original Research Article

We introduce a new “product expansion” for finite semigroups, which can easily be embedded in a “Zeiger” wreath product. We then modify the product expansion of Cω(S) (a computable collection of pointlike subsets of S) to define a relation from S to an aperiodic semigroup that computes Cω(S), proving that Cω(S) = PlA(S) (the abstractly defined collection of sets “pointlike with respect to relations into the pseudo variety A of aperiodic semigroups”). The construction given greatly simplifies the proof given in the article “Pointlike sets …” by the author [1], and (in the finite case) simplifies the arguments of the article “Prime decomposition theorem …” by Henckell et al. [2].