Ash's type II theorem, profinite topology and Malcev products

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

Two Techniques in the Area of the Star Problem

This paper deals with decision problems related to the star problem in trace monoids, which means to determine whether the iteration of a recognizable trace language is recognizable. Due to a theorem by G. Richomme from 1994 [32, 33], we know that the star problem is decidable in trace monoids which do not contain a submonoid of the form {a,c}* x {b,d}*. Here, we consider a more general problem: Is it decidable whether for some recognizable trace language and some recognizable or finite trace language P the intersection R ∩ P* is recognizable? If P is recognizable, then we show that this problem is decidale iff the underlying trace monoid does not contain a submonoid of the form {a,c}* x b*. In the case of finite languages P, we show several decidability and undecidability results

Automatic Semigroups vs Automaton Semigroups

We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup, namely the semigroup generated by a Mealy automaton encoding the behaviour of such a language of greedy normal forms under one-sided multiplication. The framework embraces many of the well-known classes of (automatic) semigroups: free semigroups, free commutative semigroups, trace or divisibility monoids, braid or Artin - Tits or Krammer or Garside monoids, Baumslag - Solitar semigroups, etc. Like plactic monoids or Chinese monoids, some neither left- nor right-cancellative automatic semigroups are also investigated, as well as some residually finite variations of the bicyclic monoid. It provides what appears to be the first known connection from a class of automatic semigroups to a class of automaton semigroups. It is worthwhile noting that, "being an automatic semigroup" and "being an automaton semigroup" become dual properties in a very automata-theoretical sense. Quadratic rewriting systems and associated tilings appear as the cornerstone of our construction

Tamari Lattices and the symmetric Thompson monoid

We investigate the connection between Tamari lattices and the Thompson group F, summarized in the fact that F is a group of fractions for a certain monoid F+sym whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice operations are the counterparts of the least common multiple and greatest common divisor operations in F+sym. As an application, we show that, for every n, there exists a length l chain in the nth Tamari lattice whose endpoints are at distance at most 12l/n.Comment: 35page

On the Complexity of Computing the Profinite Closure of a Rational Language

International audienceThe profinite topology is used in rational languages classification. In particular, several important decidability problems, related to the Malcev product, reduce to the computation of the closure of a rational language in the profinite topology. It is known that given a rational language by a deterministic automaton, computing a deterministic automaton accepting its profinite closure can be done with an exponential upper bound. This paper is dedicated the study of a lower bound for this problem: we prove that in some cases, if the alphabet contains at least three letters, it requires an exponential time

Noncommutative localization in noncommutative geometry

The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of ``spaces'', locally described by noncommutative rings and their categories of one-sided modules. We present the basics of Ore localization of rings and modules in much detail. Common practical techniques are studied as well. We also describe a counterexample for a folklore test principle. Localization in negatively filtered rings arising in deformation theory is presented. A new notion of the differential Ore condition is introduced in the study of localization of differential calculi. To aid the geometrical viewpoint, localization is studied with emphasis on descent formalism, flatness, abelian categories of quasicoherent sheaves and generalizations, and natural pairs of adjoint functors for sheaf and module categories. The key motivational theorems from the seminal works of Gabriel on localization, abelian categories and schemes are quoted without proof, as well as the related statements of Popescu, Watts, Deligne and Rosenberg. The Cohn universal localization does not have good flatness properties, but it is determined by the localization map already at the ring level. Cohn localization is here related to the quasideterminants of Gelfand and Retakh; and this may help understanding both subjects.Comment: 93 pages; (including index: use makeindex); introductory survey, but with few smaller new result