Inverse Semigroups Research Articles

On Some Classes of Regular Semigroups

In this paper we study and analyze the relationships between some special classes of regular semigroups, namely: Generalized inverse semigroups, Orthodox semigroups and locally inverse semigroups based on their idempotent properties. We then use an example to prove that every locally inverse semigroup is not a generalized inverse semigroup, thus validating the statement; “A generalized inverse semigroup has the property of being locally inverse. However, not every locally inverse semigroup is a generalized inverse semigroup.”

Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups

Given any quasi-countable, in particular, any countable inverse semigroup S , we introduce a way to equip S with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on S . This allows us to unambiguously define the uniform Roe algebra of S , which we prove can be realized as a canonical crossed product of \ell^{\infty}(S) and S . We relate these metrics to the analogous metrics on Hausdorff étale groupoids. Using this setting, we study those inverse semigroups with asymptotic dimension 0 . Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by S being locally \mathcal{L} -finite, and equivalently sparse as a metric space.

On some problems of conjugacy in semigroups

The conjugacy relation is an important tool in the study of group theory and now has been generalized to semigroups in several methods. In particular, the p-conjugacy, o-conjugacy, c-conjugacy, i-conjugacy, n-conjugacy and r-conjugacy in semigroups are introduced and investigated extensively and some problems related to these conjugacy relations are proposed. The aim of this paper is to continue the study of conjugacy relations in semigroups around some of these problems. We first point out that p-conjugacy in a semigroup that is embeddable in a group, is not necessarily transitive. Then we obtain a simple sufficient condition under which the o-conjugacy is the universal relation and prove that this condition is also necessary in some natural classes of semigroups. Finally, we explore some properties of r-conjugacy in restriction semigroups, which extends some results on i-conjugacy in inverse semigroups. Our results have answered or partially answered some problems raised by Araújo, Kinyon, Konieczny and Malheiro in [Proc.Roy.Soc.Edinburgh Sect.A 147, 1169–1214, 2017] and [J.Algebra 533, 142–173, 2019].

Finiteness properties of locally defined groups

Let X be a set and let S be an inverse semigroup of partial bijections of X . Thus, an element of S is a bijection between two subsets of X , and the set S is required to be closed under the operations of taking inverses and compositions of functions. We define \Gamma_{S} to be the set of self-bijections of X in which each \gamma \in \Gamma_{S} is expressible as a union of finitely many members of S . This set is a group with respect to composition. The groups \Gamma_{S} form a class containing numerous widely studied groups, such as Thompson’s group V , the Nekrashevych–Röver groups, Houghton’s groups, and the Brin–Thompson groups nV , among many others. We offer a unified construction of geometric models for \Gamma_{S} and a general framework for studying the finiteness properties of these groups.

RECURRENCE SETS FOR PARTIAL INVERSE

Two types of recurrence sets are introduced for inverse semigroup partial actions in topological spaces. Some applications to the basic dynamical properties of the action are indicated, covering topics as topological transitivity, limit points and periodic points. We then explore the connections of these recurrence sets with similar notions for related types of imperfect symmetries (prefix inverse semigroup expansions, partial group and groupoid actions).

Partial metrics and normed inverse semigroups

Partial metrics and normed inverse semigroups

Groups and their coset monoids

This paper studies groups and their coset monoids. The anti-abnormal subgroups of a group are firstly introduced and investigated. It is shown that a group is an N ˜ -group if and only if each subgroup of it is anti-abnormal. Also, it is proved that the coset semigroups of N ˜ -groups are exactly the E-reflexive inverse semigroups which are factorisable and the natural connection between their semilattice of idempotents and lattice of subgroups of their group of units is a dual isomorphism. Finally, some characterizations of the coset semigroups of S-groups (U-groups and residually central groups respectively) are given. This extends McAlister’s result in 1980.

The determinant of semigroups of the pseudovariety ECOM

The purpose of this paper is to compute the nonzero semigroup determinant of the class of finite semigroups in which every two idempotents commute. This class strictly contains the class of finite semigroups that have central idempotents and the class of finite inverse semigroups. This computation holds significance in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.

Formations and i-Fitting classes of inverse semigroups, congruences and languages

Formations and i-Fitting classes of inverse semigroups, congruences and languages

Inverse semigroup from metrics on doubles, III: Commutativity and (in)finiteness of idempotents

Inverse semigroup from metrics on doubles, III: Commutativity and (in)finiteness of idempotents

Properties of congruence lattices of graph inverse semigroups

From any directed graph [Formula: see text] one can construct the graph inverse semigroup [Formula: see text], whose elements, roughly speaking, correspond to paths in [Formula: see text]. Wang and Luo showed that the congruence lattice [Formula: see text] of [Formula: see text] is upper-semimodular for every graph [Formula: see text], but can fail to be lower-semimodular for some [Formula: see text]. We provide a simple characterization of the graphs [Formula: see text] for which [Formula: see text] is lower-semimodular. We also describe those [Formula: see text] such that [Formula: see text] is atomistic, and characterize the minimal generating sets for [Formula: see text] when [Formula: see text] is finite and simple.

Restrictions on local embeddability into finite semigroups

We expand the concept of local embeddability into finite structures (LEF) for the class of semigroups with investigations of non-LEF structures, a closely related generalising property of local wrapping of finite structures (LWF) and inverse semigroups. The established results include a description of a family of non-LEF semigroups unifying the bicyclic monoid and Baumslag–Solitar groups and demonstrating that inverse LWF semigroups with finite number of idempotents are LEF.

On amenable-like properties of some Banach algebras with applications to study their nilpotent ideals

In this current article, we show that a non-zero closed nilpotent ideal in a Johnson pseudo-contractible Banach algebra cannot be [Formula: see text]-predual. We prove also that a nilpotent ideal in a Johnson pseudo-contractible Banach algebras with the approximation property is forced to be zero. Among other things, we characterize the property [Formula: see text] of some semigroup algebras associated with inverse semigroups. More especially, for an inverse semigroup [Formula: see text] such that [Formula: see text] is uniformly locally finite, [Formula: see text] has the property [Formula: see text] if and only if each maximal subgroup of [Formula: see text] is amenable, where [Formula: see text] is the set of idempotents of [Formula: see text]. Furthermore, we study the property [Formula: see text] of some [Formula: see text]-Lau product structures.

Some Propositions About Inverse Semigroups

Inverse semigroups are an important subclass of the class of semigroups. An important role in the properties of the semigroups in general and in the class of inverse semigroups in particular is played by the set of their idempotents, which we denote by E or ES for a semigroup S. In the first part of this paper we will show some propositions about left unitary, right unitary, unitary subsets and closed subsets in the inverse semigroups. We will also see some properties of the maximum idempotent - separating congruence in an inverse semigroup S and its connection with Clifford’s semigroups.
 
 Received: 19 February 2024 / Accepted: 15 March 2024 / Published: 23 March 2024

Monogenic free inverse semigroups and partial automorphisms of regular rooted trees

For a one-to-one partial mapping on an infinite set, we present a criterion in terms of its cycle-chain decomposition that the inverse subsemigroup generated by this mapping is monogenic free inverse.
 We also give a sufficient condition for a regular rooted tree partial automorphism to extend to a partial automorphism of another regular rooted tree so that the inverse semigroup gene\-ra\-ted by this extended partial automorphism is monogenic free inverse. The extension procedure we develop is then applied to $n$-ary adding machines.

Algebraic Properties of the Semigroup of Partial Isometries of a Finite Chain

This work analyses the algebraic properties of the sub-semi group of partial Isometries (〖DP〗_n) and of order preserving partial isometries (〖ODP〗_n) of a finite chain X_n={1,2,…n}, with a symmetric inverse semigroup I_n defined on it. It also investigates the subsemigroups 〖DP〗_n and 〖ODP〗_n for cycle structure and shows that 〖ODP〗_n is a 0 -E- unitary inverse semigroup.

Fuzzy admissible congruences on type B semigroups with E-properties, and their applications

Type B semigroups are described as the generalized inverse semigroups in the range of abundant semigroup. Motivated by studying fuzzy congruences in inverse semigroups, and as a continuation of N. Kuroki’s work in inverse semigroups and our work in abundant semigroups in terms of fuzzy subsets, this paper considers fuzzy admissible congruences on some classes of type B semigroups. Our main purpose is to show when a fuzzy admissible congruence on a type B semigroup with E-properties is E-properties preserving. In particular, we get some sufficient and necessary conditions for some classes of type B semigroups to be primitive, E-unitary and E-reflexive, respectively. As an application, we extend our results to the cases of inverse semigroups.

A groupoid approach to regular ⁎-semigroups

In this paper we develop a new groupoid-based structure theory for the class of regular ⁎-semigroups. This class occupies something of a ‘sweet spot’ between the important classes of inverse and regular semigroups, and contains many natural examples. Some of the most significant families include the partition, Brauer and Temperley-Lieb monoids, among other diagram monoids.Our main result is that the category of regular ⁎-semigroups is isomorphic to the category of so-called ‘chained projection groupoids’. Such a groupoid is in fact a triple (P,G,ε), where:•P is a projection algebra (in the sense of Imaoka and Jones),•G is an ordered groupoid with object set P, and•ε:C→G is a special functor, where C is a certain natural ‘chain groupoid’ constructed from P.Roughly speaking: the groupoid G=G(S) remembers only the ‘easy’ products in a regular ⁎-semigroup S; the projection algebra P=P(S) remembers only the ‘conjugation action’ of the projections of S; and the functor ε=ε(S) tells us how G and P ‘fit together’ in order to recover the entire structure of S. In this way, we obtain the first completely general structure theorem for regular ⁎-semigroups.As a consequence of our main result, we give a new proof of the celebrated Ehresmann–Schein–Nambooripad Theorem, which establishes an isomorphism between the categories of inverse semigroups and inductive groupoids. Other applications will be given in future works.We consider several examples along the way, and pose a number of problems that we believe are worthy of further attention.

Orderability of the prefix expansion of an ordered inverse semigroup

Orderability of the prefix expansion of an ordered inverse semigroup

Broken Möbius categories of $$Q_{3}$$-type and their split inverse semigroups

Broken Möbius categories of $$Q_{3}$$-type and their split inverse semigroups