E-unitary Inverse Semigroups Research Articles

The uniform Roe algebra of an inverse semigroup

Given a discrete and countable inverse semigroup S one can study, in analogy to the group case, its geometric aspects. In particular, we can equip S with a natural metric, given by the path metric in the disjoint union of its Schützenberger graphs. This graph, which we denote by ΛS, inherits much of the structure of S. In this article we compare the C*-algebra RS, generated by the left regular representation of S on ℓ2(S) and ℓ∞(S), with the uniform Roe algebra over the metric space, namely Cu⁎(ΛS). This yields a characterization of when RS=Cu⁎(ΛS), which generalizes finite generation of S. We have termed this by admitting a finite labeling (or being FL), since it holds when ΛS can be labeled in a finitary manner.The graph ΛS, and the FL condition, also allow to analyze large scale properties of ΛS and relate them with C*-properties of the uniform Roe algebra. In particular, we show that domain measurability of S (a notion generalizing Day's definition of amenability of a semigroup, cf., [6]) is a quasi-isometric invariant of ΛS. Moreover, we characterize property A of ΛS (or of its components) in terms of the nuclearity and exactness of the corresponding C*-algebras. We also treat the special classes of F-inverse and E-unitary inverse semigroups from this large scale point of view.

Aleph _0-categoricity of semigroups

In this paper we initiate the study of aleph _0-categorical semigroups, where a countable semigroup S is aleph _0-categorical if, for any natural number n, the action of its group of automorphisms {text {Aut}}(S) on S^n has only finitely many orbits. We show that aleph _0-categoricity transfers to certain important substructures such as maximal subgroups and principal factors. We examine the relationship between aleph _0-categoricity and a number of semigroup and monoid constructions, namely Brandt semigroups, direct sums, 0-direct unions, semidirect products and {mathcal {P}}-semigroups. As a corollary, we determine the aleph _0-categoricity of an E-unitary inverse semigroup with finite semilattice of idempotents in terms of that of the maximal group homomorphic image.

A unifying approach to the Margolis–Meakin and Birget–Rhodes group expansion

Let G be a group. We show that the Birget–Rhodes prefix expansion $$G^{Pr}$$ and the Margolis–Meakin expansion M(X; f) of G with respect to $$f:X\rightarrow G$$ can be regarded as inverse subsemigroups of a common E-unitary inverse semigroup P. We construct P as an inverse subsemigroup of an E-unitary inverse monoid $$U/\zeta $$ which is a homomorphic image of the free product U of the free semigroup $$X^+$$ on X and G. The semigroup P satisfies a universal property with respect to homomorphisms into the permissible hull C(S) of a suitable E-unitary inverse semigroup S, with $$S/\sigma _S=G$$ , from which the characterizing universal properties of $$G^{Pr}$$ and M(X; f) can be recaptured easily.

On E-unitary inverse and related semigroups

In this paper, we study the structure of E-unitary inverse semigroups together with related classes such as proper left ample semigroups and proper left restriction semigroups. Important theorems on proper left ample semigroups and proper left restriction semigroups analogous to the P-theorem for E-unitary inverse semigroups are considered. From these theorems, we obtain a structure theorem for proper inverse semigroups.

On inverse semigroup $C^*$-algebras and crossed products

We describe the C � -algebra of an E-unitary or strongly 0- E-unitary inverse semigroup as the partial crossed product of a commu- tative C � -algebra by the maximal group image of the inverse semigroup. We give a similar result for the C � -algebra of the tight groupoid of an inverse semigroup. We also study conditions on a groupoid C � -algebra to be Morita equivalent to a full crossed product of a commutative C � - algebra with an inverse semigroup, generalizing results of Khoshkam and Skandalis for crossed products with groups.

Amenability of the Restricted Fourier Algebras

We discuss amenability of the restricted Fourier-Stieltjes algebras on inverse semigroups. We show that, for an E-unitary inverse semigroup, amenability of the restricted Fourier-Stieltjes algebra is related to the amenability of an associated Banach algebra on a Fell bundle.

The Universal Covering of an Inverse Semigroup

We examine an inverse semigroup T in terms of the universal locally constant covering of its classifying topos Open image in new window. In particular, we prove that the fundamental group of Open image in new window coincides with the maximum group image of T. We explain the connection between E-unitary inverse semigroups and locally decidable toposes, characterize E-unitary inverse semigroups in terms of a kind of geometric morphism called a spread, characterize F-inverse semigroups, and interpret McAlister’s “P-theorem” in terms of the universal covering.

Semigroups and Toposes

We establish a semigroup presentation of the classifying topos of an inverse semigroup. For this purpose we consider a category larger than inverse semigroups and prehomomorphisms, whose objects we call left *-semigroups. We show that any etendue has a left *-semigroup presentation. We show that any left *-semigroup may be codensely embedded in an inverse semigroup. We also establish a strictly topos description of E-unitary inverse semigroups.

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 Exel's universal inverse semigroup [Formula: see text], which has the property that partial actions of the group G give rise to actions of the inverse semigroup [Formula: see text]. 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 uniform word problem for groups and finite Rees quotients of E-unitary inverse semigroups

If C is a class of groups closed under taking subgroups, we show that the decidability of the uniform word problem for C is implied by the decidability of the membership problem for the class of finite Rees quotients of E-unitary inverse semigroups with maximal group image in C . The converse is shown if C is a pseudovariety. When C is a pseudovariety, the above problems are shown to be equivalent to the problem of embedding a finite labeled graph in the Cayley graph of a group in C . This latter problem is shown to be equivalent to deciding whether a finite labeled graph is a Schützenberger graph of an E-unitary inverse semigroup with maximal group image in C .

A syntactic approach to covers forE‐dense semigroups over group varieties

A major result of D. B. McAlister is that every inverse semigroup is an idempotent separating morphic image of an E-unitary inverse semigroup. The result has been generalized by various authors (including Szendrei, Takizawa, Trotter, Fountain, Almeida, Pin, Weil) to any semigroup of the following types: orthodox, regular, ii-dense with commuting idempotents, E-dense with idempotents forming a subsemigroup, and is-dense. In each case, a semigroup is a morphic image of a semigroup in which the weakly self conjugate core is unitary and separated by the homomorphism. In the present paper, for any variety H of groups and any E-dense semigroup S, the concept of an “H-verbal subsemigroup” of S is introduced which is intimately connected with the least H-congruence on S. What is more, this construction provides a short and easy access to covering results of the aforementioned kind. Moreover, the results are generalized, in that covers over arbitrary group varieties are constructed for any E-dense semigroup. If the given semigroup enjoys a “regularity condition” such as being eventually regular, group bound, or regular, then so does the cover.

PROPER WEAKLY LEFT AMPLE SEMIGROUPS

Much of the structure theory of inverse semigroups is based on constructing arbitrary inverse semigroups from groups and semilattices. It is known that E-unitary (or proper) inverse semigroups may be described as P-semigroups (McAlister), or inverse subsemigroups of semidirect products of a semilattice by a group (O'Carroll) or Cu-semigroups built over an inverse category acted upon by a group (Margolis and Pin). On the other hand, every inverse semigroup is known to have an E-unitary inverse cover (McAlister). The aim of this paper is to develop a similar theory for proper weakly left ample semigroups, a class with properties echoing those of inverse semigroups. We show how the structure of semigroups in this class is based on constructing semigroups from unipotent monoids and semilattices. The results corresponding to those of McAlister, O'Carroll and Margolis and Pin are obtained.

HNN EXTENSIONS OF SEMILATTICES

The main purpose of this paper is to investigate properties of an HNN extension of a semilattice, to give its equivalent characterizations and to discuss similarities with free groups. An HNN extension of a semilattice is shown to be a universal object in a certain category and an F-inverse cover over a free group for every inverse semigroup in the category. We also show that a graph with respect to a certain subset of an HNN extension of a semilattice is a tree and that this property characterizes an HNN extension of a semilattice. Moreover, we look into three subclasses: the class of full HNN extensions of semilattices with an identity, the class of universally E-unitary inverse semigroups and the class of HNN extensions of finite semilattices. The first class consists of factorizable E-unitary inverse semigroups whose maximal group homomorphic images are free. We obtain a generalization of the Nielsen–Schreier subgroup theorem to this class. The second consists of inverse semigroups presented by relations on Dyck words. An inverse semigroup in the third class has a relatively easy finite presentation using a Dyck language and has solvable word problem.

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 E-unitary inverse semigroups to those 0-E-unitary inverse semigroups which admit a 0-restricted, idempotent pure prehomomorphism to a primitive inverse semigroup. In this paper, we concentrate on finding necessary and sufficient conditions for the existence of such prehomomorphisms in the case of 0-E-unitary inverse monoids. A class of inverse monoids which satisfy our conditions automatically are those which are unambiguous except at zero, such as the polycyclic monoids.

Inverse semigroups with zero: covers and their structure

AbstractWe obtain analogues, in the setting of semigroups with zero, of McAlister's convering theoren and the structure theorems of McAlister, O'Carroll, and Margolis and Pin. The covers come from a classCof semigroups defined by modifying one of the many characterisations ofE-unitary inverse semigroups, namely, that an inverse semigroups isE-unitary if and only if it is an inverse image of an idempotent-pure homomorphism onto a group. The classCis properly contained in the class of allE*-unitary inverse semigroups introduced by Szendrei but properly contains the class of strongly categoricalE*-unitary semigroups recently considered by Gomes and Howie.

An Application of Groupoids of Fractions to Inverse Semigroups

We describe an application of category theory to the theory of inverse semigroups: we prove the P-theorem for E-unitary inverse semigroups using groupoids of fractions of their associated division categories.

Idempotent pure extensions by inverse semigroups via quivers

In this paper we describe idempotent pure regular extensions by inverse semigroups by means of quivers and actions of inverse semigroups, generalising the category and action of groups approach presented by Margolis and Pin for E-unitary inverse semigroups. By making use of these new tools, we can uniformly reprove O'Carroll's and Billhardt's characterisations of idempotent pure inverse extensions by inverse semigroups as L m -semigroups and as inverse subsemigroups of a λ-semidirect product of a semilattice by an inverse semigroup, respectively.

Trace functions on the algebras of certain E-unitary inverse semigroups

A construction is given for a trace function on the semigroup algebra of a certain type of E-unitary inverse semigroup over any subfield of the complex field that is closed under complex conjugation. In particular, the method applies to the semigroup algebras of free inverse semigroups of arbitrary rank.

Normal-convex embeddings of inverse semigroups

Normal-convex embeddings are introduced for inverse semigroups, generalizing the group-theoretic concept, due to Papakyriakopoulos [4]. It is shown that every E-unitary inverse semigroup admits a normal-convex embedding into a semidirect product of a semilattice by a group, a stronger version of a result by O'Carroll [3]. A general embedding result for inverse semigroups is also obtained.

The geometric theory of inverse semigroups I: E-unitary inverse semigroups

We show how the correspondence between inverse semigroups and inductive groupoids (a class of ordered groupoids) leads to a reinterpretation of McAlister's P-Theorem as a function extension theorem. This result can only be expressed, in general, by working within a larger category: namely that of functorially ordered groupoids. The consequences of this approach for the theory of E-unitary covers of inverse semigroups and its generalisations are worked out in subsequent papers.