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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].

This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every element has at most one semigroup inverse. As a consequence of the cover conjecture - also verified by Ash - it follows that block groups are precisely the divisors of power monoids of finite groups. The proof of this last fact uses earlier results of the authors and the deepest tools and results from global semigroup theory. We next give connections with the profinite group topologies on finitely generated free monoids and free groups. In particular, we show that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for the free group). Now Ash's theorem implies that the two topological conjectures are true and independently, a direct proof of the topological conjecture for the free group has been recently obtained by Ribes and Zalesskii. An important consequence is that a rational subset of a finitely generated free group G is closed in the profinite topology if and only if it is a finite union of sets of the form gH1H2... Hn, where each Hi is a finitely generated subgroup of G. This significantly extends classical results by M. Hall. Finally we return to the roots of this problem and give connections with the complexity theory of finite semigroups. We show that the largest local complexity function in the sense of Rhodes and Tilson is computable.