Mark V. Lawson

Inverse semigroups : the theory of partial symmetries

Introduction to inverse semigroups elementary properties of inverse semigroups Ehresmanns maximum enlargement theorem presentations of inverse monoids inverse semigroups and formal languages the type II theorem complements.

Semigroups and Ordered Categories. I. The Reduced Case

Abstract We investigate the relationship which exists between certain classes of ordered small categories, introduced by Charles Ehresmann in the course of his work on local structures, and the class of U -semiabundant semigroups, first studied by El-Qallali and by de Barros. U -semiabundant semigroups embrace both the abundant and the regular semigroups—the former have been studied by Fountain and his students in a number of papers, the latter constitute a well-known and wellinvestigated branch of semigroup theory. One consequence of the ideas presented in this paper is that the work of Nambooripad on inductive groupoids and regular semigroups should not be seen as sui generis but as belonging to the general framework of the theory of ordered small categories.

PARTIAL ACTIONS OF GROUPS

A partial action of a group G on a set X is a weakening of the usual notion of a group action: the function G×X→X that defines a group action is replaced by a partial function; in addition, the existence of g·(h·x) implies the existence of (gh)·x, but not necessarily conversely. Such partial actions are extremely widespread in mathematics, and the main aim of this paper is to prove two basic results concerning them. First, we obtain an explicit description of Exels universal inverse semigroup , which has the property that partial actions of the group G give rise to actions of the inverse semigroup . We apply this result to the theory of graph immersions. Second, we prove that each partial group action is the restriction of a universal global group action. We describe some applications of this result to group theory and the theory of E-unitary inverse semigroups.

The natural partial order on an abundant semigroup

In this paper we will study the properties of a natural partial order which may be defined on an arbitrary abundant semigroup: in the case of regular semigroups we recapture the order introduced by Nambooripad [24]. For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1, 2] and further studied by Chacron [7]. Burmistrovic [6] investigated Sussmans order on separative semigroups. In the abundant case his order coincides with ours: some order theoretic properties of such semigroups may be found in a paper by Burgess [5]. Many properties and constructions on abundant semigroups may be described in terms of its natural partial order: one of the main results of Section 2 is the connection we establish between idempotent connectedness and the partial order being, in some sense, self-dual. In Section 3 we extend Nambooripads Theorem 3.3 [24] and show that the order is compatible with the multiplication on a concordant semigroup just when the semigroup is locally type A. In Section 4 we obtain a description of the finest 0restricted primitive good congruence on a concordant semigroup. Section 1 is a preliminary section in which we consider, in particular, the behaviour of good homomorphisms between abundant semigroups and obtain a generalisation of Lallements Lemma.

REES MATRIX SEMIGROUPS

In this paper we provide a new, abstract characterisation of classical Rees matrix semigroups over monoids with zero. The corresponding abstract class of semigroups is obtained by abstracting a number of algebraic properties from completely 0-simple semigroups: in particular, the relationship between arbitrary elements and idempotents.

NON-COMMUTATIVE STONE DUALITY: INVERSE SEMIGROUPS, TOPOLOGICAL GROUPOIDS AND C*-ALGEBRAS

We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse ∧-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse ∧-semigroups arise as completions of inverse ∧-semigroups we call pre-Boolean. An inverse ∧-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where tight filters are defined by combining ideas of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson–Higman groups Gn, r. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz–Krieger C*-algebras. An elementary application of our theory shows that the finite, fundamental Boolean inverse ∧-semigroups are just the finite direct products of finite symmetric inverse monoids. Finally, we explain how tight filters are related to prime filters setting the scene for future work.

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.

Orthogonal completions of the polycyclic monoids

We introduce the notion of an orthogonal completion of an inverse monoid with zero. We show that the orthogonal completion of the polycyclic monoid on n generators is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated right ideals of the free monoid on n generators, and so we can make a direct connection with the Thompson groups Vn,1.

A noncommutative generalization of stone duality

We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The group of units of Cn is the Thompson group Vn,1.

The Structure of 0- E -unitary inverse semigroups I: the monoid case

This is the first of three papers in which we generalise the classical McAlister structure theory for £-unitaryinverse semigroups to those O-E-unitary inverse semigroups which admit a O-restricted, idempotent pureprehomomorphism to a primitive inverse semigroup. In this paper, we concentrate on finding necessary andsufficient conditions for the existence of such prehomomorphisms in the case of 0-£-unitary inverse monoids.A class of inverse monoids which satisfy our conditions automatically are those which are unambiguousexcept at zero, such as the polycyclic monoids.1991 Mathematics subject classification: 20M18 (18B40).