Ash's Type Ii Theorem, Profinite Topology and Malcev Products: Part I

Abstract

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 reviewing 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 g ∈ G and each Hi is a finitely generated subgroup of G. This significantly extends classical results of 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.

Abelian Kernels of Monoids of Order-preserving Maps and of Some of Its Extensions

“…The notion of H-kernel is tightly related to an important operator of pseudovarieties: the Mal'cev product (see [12]). Its definition, when the first factor is a pseudovariety V of monoids and the second factor is a pseudovariety H of groups, may be given as follows:…”

“…It survived as a conjecture almost twenty years and became a theorem after independent and deep work of Ash [2] and Ribes and Zalesskiȋ [17]. For history, motivation and consequences of the conjecture we refer the reader to [12].…”

Semidirect Products of Ordered Semigroups

“…The decomposition of block-groups has been thoroughly studied (see [11,13,14] or [10,18] for a survey). Let PG be the variety of monoids generated by all monoids of the form P (G), where G is a group and let BG be the variety of block-groups.…”

Closures of regular languages for profinite topologies

“…In an equivalent form [10,11], Ash's theorem was rediscovered by Herwig and Lascar [25]. The stronger conjecture, equivalent to the Rhodes type II conjecture [23], was also proved by Ribes and Zalesskiȋ [34]. An elementary and constructive proof was obtained recently by Auinger [18] (see also [19]).…”