Stuart W. Margolis

ASH'S TYPE II THEOREM, PROFINITE TOPOLOGY AND MALCEV PRODUCTS: PART I

This paper is concerned with the many deep and far reaching consequences of Ashs 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 Ashs 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.

Representation theory of finite semigroups, semigroup radicals and formal language theory

In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and Cernys conjecture for an important class of automata.

E-unitary inverse monoids and the Cayley graph of a group presentation

Abstract Geometric methods have played a fundamental and crucial role in combinatorial group theory almost from the inception of that field. In this paper we initiate a study of the use of some of these methods in inverse semigroup theory. We modify a lemma of I. Simon and show how to construct E -unitary inverse monoids from the free idempotent and commutative category over the Cayley graph of the maximal group image. The construction provides an expansion from the category of X -generated groups to the category of X -generated E -unitary inverse monoids and specializes to a construction of certain relatively free E -unitary inverse monoids. We show more generally that this construction is the left adjoint of the maximal group image functor. Munns solution to the word problem for the free inverse monoids and several of the results of McAlister and McFadden on the free inverse semigroup with two commuting generators may be obtained fairly easily from the construction. We construct the free product of E -unitary inverse monoids, thus providing an alternate construction to that of Jones.

FREE INVERSE MONOIDS AND GRAPH IMMERSIONS

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

Inverse monoids, trees and context-free languages

This paper is concerned with a study of inverse monoids presented by a set X subject to relations of the form e i = f i , i ∈ I, where e i and f i are Dyck words, i.e. idempotents of the free inverse monoid on X. Some general results of Stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the Cayley graph of the free group on X. In the finitely presented case the word problem is solved by using Rabins theorem on the second order monadic logic of the infinite binary tree. Some connections with the theory of rational subsets of the free group and the theory of context-free languages are explored

Algorithmic problems for finite groups and finite 0-simple semigroups

Abstract It is shown that the embeddability of a finite 4-nilpotent semigroup into a 0-simple finite semigroup with maximal groups from a pseudovariety V is decidable if and only if the universal theory of the class V is decidable. We show that it is impossible to replace 4 by 3 in this statement. We also show that if the membership in V is decidable then the membership in the pseudovariety generated by the class of all finite 0-simple semigroups with subgroups from V is decidable while the membership in the quasi-variety generated by this class of 0-simple semigroups may be undecidable.

Expansions of inverse semigroups

We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms which we term F-morphisms. These play the same role in the theory of idempotent-pure homomorphisms that F-inverse monoids play in the theory of E-unitary inverse semigroups.

Words guaranteeing minimal image

Given a positive integer n and a finite alphabet A, a word w over A is said to guarantee minimal image if, for every homomorphism f from the free monoid A* over A into the monoid of all transformations of an n-element set, the range of the transformation wf has the minimum cardinality among the ranges of all transformations of the form vf where v runs over A*. Although the existence of words guaranteeing minimal image is pretty obvious, the problem of their explicit description is very far from being trivial. Sauer and Stone in 1991 gave a recursive construction for such a word w but the length of the word resulting from that construction was doubly exponential (as a function of n). We first show that some known results of automata theory immediately lead to an alternative construction which yields a simpler word that guarantees minimal image: it has exponential length, more precisely, its length is O(|A|^(n^3-n)). Then using a different approach, we find a word guaranteeing minimal image similar to that of Sauer and Stone but of the length O(|A|^(n^2-n)). On the other hand, we observe that the length of any word guaranteeing minimal image cannot be less than |A|^(n-1).

Quivers of monoids with basic algebras

We compute the quiver of any nite monoid that has a basic algebra over an algebraically closed eld of characteristic zero. More generally, we reduce the computation of the quiver over a splitting eld of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as DO) to representation-theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of R-trivial monoids, we also provide a semigroup-theoretic description of the projective indecomposable modules and compute the Cartan matrix.

Languages and Inverse Semigroups

We show how several famous combinatorial sequences appear in the context of nilpotent elements of the full symmetric inverse semigroup I Sn. These sequences appear either as cardinalities of certain nilpotent subsemigroups or as the numbers of special nilpotent elements and include the Lah numbers, the Bell numbers, the Stirling numbers of the second kind, the binomial coefficients and the Catalan numbers.