Class Of Inverse Semigroups Research Articles

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

Semigroups of straight left inverse quotients

Let Q be an inverse semigroup. A subsemigroup S of Q is a left I-order in Q and Q is a semigroup of left I-quotients of S if every element in Q can be written as a^{-1}b, where a, b in S and a^{-1} is the inverse of a in the sense of inverse semigroup theory. If we insist on being able to take a and b to be mathscr {R}-related in Q we say that S is straight in Q and Q is a semigroup of straight left I-quotients of S. We give a set of necessary and sufficient conditions for a semigroup to be a straight left I-order. The conditions are in terms of two binary relations, corresponding to the potential restrictions of {mathscr {R}} and {mathscr {L}} from an oversemigroup, and an associated partial order. Our approach relies on the meet structure of the mathscr {L}-classes of inverse semigroups. We prove that every finite left I-order is straight and give an example of a left I-order which is not straight.

Algebras defined by equations

We show that an elementary class of algebras is closed under the taking of homomorphic images and direct products if and only if the class consists of all algebras that satisfy a set of (generally simultaneous) equations. For classes of regular semigroups this allows an interpretation of a universal algebraic nature that is formulated entirely in terms of the associative binary operation of the semigroup, which serves as an alternative to the approach via so called e-varieties. In particular we prove that classes of Inverse semigroups, Orthodox semigroups, and E-solid semigroups are equational in our sense. We also examine which equations are valid in every semigroup.

Amenability and paradoxicality in semigroups and C⁎-algebras

We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence.In the context of inverse semigroups S we give a characterization of invariant measures on S (in the sense of Day) in terms of two notions: domain measurability and localization. Given a unital representation of S in terms of partial bijections on some set X we define a natural generalization of the uniform Roe algebra of a group, which we denote by RX. We show that the following notions are then equivalent: (1) X is domain measurable; (2) X is not paradoxical; (3) X satisfies the domain Følner condition; (4) there is an algebraically amenable dense*-subalgebra of RX; (5) RX has an amenable trace; (6) RX is not properly infinite and (7) [0]≠[1] in the K0-group of RX. We also show that any tracial state on RX is amenable. Moreover, taking into account the localization condition, we give several C*-algebraic characterizations of the amenability of X. Finally, we show that for a certain class of inverse semigroups, the quasidiagonality of Cr⁎(X) implies the amenability of X. The reverse implication (which is a natural generalization of Rosenberg's conjecture to this context) is false.

Conjugacy in inverse semigroups

In a group G, elements a and b are conjugate if there exists g∈G such that g−1ag=b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a∼ib, if there exists g∈S1 such that g−1ag=b and gbg−1=a. The purpose of this paper is to study the conjugacy ∼i in several classes of inverse semigroups: symmetric inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid, stable inverse semigroups, and free inverse semigroups.

On generalized Ehresmann semigroups

Abstract As a generalization of the class of inverse semigroups, the class of Ehresmann semigroups is introduced by Lawson and investigated by many authors extensively in the literature. In particular, Gomes and Gould construct a fundamental Ehresmann semigroup CE from a semilattice E which plays for Ehresmann semigroups the role that TE plays for inverse semigroups, where TE is the Munn semigroup of a semilattice E. From a varietal perspective, Ehresmann semigroups are derived from reduction of inverse semigroups. In this paper, from varietal perspective Ehresmann semigroups are extended to generalized Ehresmann semigroups derived instead from normal orthodox semigroups (i.e. regular semigroups whose idempotents form normal bands) with an inverse transversal. We present here a semigroup C(I,Λ,E∘) from an admissible triple (I, Λ, E∘) that plays for generalized Ehresmann semigroups the role that CE from a semilattice E plays for Ehresmann semigroups. More precisely, we show that a semigroup is a fundamental generalized Ehresmann semigroup whose admissible triple is isomorphic to (I, Λ, E∘) if and only if it is (2,1,1,1)-isomorphic to a quasi-full (2,1,1,1)-subalgebra of C(I,Λ,E∘). Our results generalize and enrich some results of Fountain, Gomes and Gould on weakly E-hedges semigroups and Ehresmann semigroups.

Amenability and uniqueness for groupoids associated with inverse semigroups

We investigate recent uniqueness theorems for reduced \(C^*\)-algebras of Hausdorff etale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full \(C^*\)-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.

Fourier Inversion for Finite Inverse Semigroups

This paper continues the study of Fourier transforms on finite inverse semigroups, with a focus on Fourier inversion theorems and FFTs for new classes of inverse semigroups. We begin by introducing four inverse semigroup generalizations of the Fourier inversion theorem for finite groups. Next, we describe a general approach to the construction of fast inverse Fourier transforms for finite inverse semigroups complementary to an approach to FFTs given in previous work. Finally, we give fast inverse Fourier transforms for the symmetric inverse monoid and its wreath product by arbitrary finite groups, as well as fast Fourier and inverse Fourier transforms for the planar rook monoid, the partial cyclic shift monoid, and the partial rotation monoid.

Eω-Clifford congruences and Eω-E-reflexive congruences on an inverse semigroup

This paper enlarges the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower T and lower K. Two classes of inverse semigroups, namely Eω-Clifford semigroups and Eω-E-reflexive semigroups, are studied. ((ωT)K)T and (((ωT)K)T)K are found to be the least Eω-Clifford and Eω-E-reflexive congruences on S, respectively. Characterizations of the congruences ((ωT)K)T and (((ωT)K)T)K are developed. The lattices of all Eω-Clifford and Eω-E-reflexive congruences are also studied.

Computer Solutions of Problems in Inverse Semigroups

In 1981, Tamura posed a number of problems regarding the axiomatic definition of inverse semigroups. The main goal of this article is to use automated reasoning to solve these problems. In the process, we find some new defining sets of identities for the class of inverse semigroups and provide a single identity for groups in terms of two binary operations.

On maximal stable orders on an inverse semigroup of finite rank with zero

We consider maximal stable orders on semigroups that belong to a certain class of inverse semigroups of finite rank.

$$\mathcal{P}\mathcal{A}{\text{-isomorphisms}}$$ of inverse semigroups

A partial automorphism of a semigroup $S$ is any isomorphism between its subsemigroups, and the set all partial automorphisms of $S$ with respect to composition is the inverse monoid called the partial automorphism monoid of $S$. Two semigroups are said to be $\cPA$-isomorphic if their partial automorphism monoids are isomorphic. A class $\K$ of semigroups is called $\cPA$-closed if it contains every semigroup $\cPA$-isomorphic to some semigroup from $\K$. Although the class of all inverse semigroups is not $\cPA$-closed, we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is $\cPA$-closed. It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is $\cPA$-closed. A semigroup is called $\cPA$-determined if it is isomorphic or anti-isomorphic to any semigroup that is $\cPA$-isomorphic to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are $\cPA$-determined.

McAlister Semigroups

This paper is the first in a sequence in which we investigate the algebraic structure of Kellendonk's one-dimensional tiling semigroups. However, as a prelude to studying the tiling semigroups themselves we study here a class of inverse semigroups which are generated by disjoint elements which we call “McAlister semigroups.” They were first introduced by Don McAlister as a by-product of his work on inverse semigroups separated over a subsemigroup.

HNN Extensions of Inverse Semigroups and Applications

Originally the concept of an HNN extension of a group was introduced by Higman, Neumann and Neumann in their study of embeddability of groups. Howie introduced the concept of HNN extensions of semigroups and showed embeddability in the case that the associated subsemigroups are unitary. On the other hand, T. E. Hall showed the embeddability of HNN extensions of inverse semigroups of a special type in his survey article on amalgamation of inverse semigroups. We introduce a more general definition of an HNN extension and show that free inverse semigroups and the bicyclic semigroup are HNN extensions of semilattices as examples of our new construction. We discuss weak HNN embeddability in several classes of semigroups and strong HNN embeddability in the class of inverse semigroups. One of our main purposes in the study of HNN extensions of inverse semigroups is to employ HNN extensions to examine some algorithmic problems. We prove the undecidability of Markov properties of finitely presented inverse semigroups using HNN extensions. This result was announced by Vazhenin in 1978, but no proof of it has been published to date. We also show undecidability of several non-Markov properties and discuss some undecidable problems on finitely generated inverse subsemigroups of finitely presented inverse semigroups.

Inverse semigroups through groups

In most introductory abstract algebra courses the topics covered are groups, rings and fields. In recent years, semigroups have been included due to their value in certain areas of computer science. However, after a few examples of familiar semigroups, the subject is forgotten and concentration is placed on developing the group structure. In this paper we attempt to show how one could continue examination of semigroups in conjunction with the development of groups. Since the group structure is such a rich and useful one, it would be natural to explore semigroups which are, in some sense, close to groups. The class of inverse semigroups fits this description.

Varieties of completely semisimple inverse semigroups

In [10] Vagner essentially described the class of inverse semigroups as a variety. Research into this class as a variety accelerated with the derivation of good descriptions of the free inverse semigroup of rank one by Gluskin [3] and by Eberhart and Selden [2] and of the free inverse semigroup of arbitrary rank by Scheiblich [9]. However, the first substantial insight into the structure of the lattice of varieties was provided by Djadchenko [1] and Kleiman [4], [5]. In particular, they showed that the lattice has three isomorphic layers at the bottom. The first consisting of varieties of groups, the second of varieties of Clifford semigroups and the third of varieties generated by Brandt semigroups. In [5], Kleiman provides a basis of identities for each of these varieties. Whereas there is a natural description of the semigroups in the first two layers referred to above, there is no such description of those appearing in the third layer. The second section of this paper is devoted to obtaining structural information about these semigroups. It is shown, for instance, that any semigroup appearing in a variety contained in f~ v ~P(B~), where f~ is the variety of groups and ~P(B2) is the variety generated by the Brandt semigroup B 2 = J/d°(1; I ; I; A) with [I1 = 2, is a subdirect product of Brandt semigroups, is completely semisimple and satisfies ~-majorization in the sense of Petrich [7]. In Section 3, four different characterizations are given of varieties of inverse semigroups every member of which is completely semisimple and in which is a congruence. Although this latter class, W~c~o~, is not itself a variety, the varieties contained in rg~9°~ form a sublattice of the lattice of all varieties and some aspects of this sublattice are considered. In particular, it is shown that the even smaller sublattice consisting of varieties contained in 7(B21) v is non-modular.

Basis properties for inverse semigroups

In a previous paper [9], the author showed that a wide class of inverse semigroups (for example free inverse semigroups, semilattices and finite combinatorial inverse semigroups) possess the strong basis property, and thus the basis property, which we now define. If S is an invcrsc semigroup and C < C; ::z S, a CT-basis for I’ is a subset S of V, minimal such that (Z! u .Y) 7: V, and a hasis for V is an O-basis for V, S is said to have the basis property [strong basis property] if for any inverse subsemigroup V of S [and inverse subsemigroup I,’ of I’] any two bases for C’ [U-bases for I’] have the same cardinality. (One notable consequence of the strong basis property for S is that for any j’-class J of S, any two U-bases for V have the same number of elements in J [9, Theorem 3.91). Two questions were put in [9]:

On a Class of Inverse Semigroups

In this note we investigate the structure of a special class of inverse sernigroups-inverse semigroups the non-zero idempotents of which are primitive. We show that an inverse semigroup S has its non-zero idempoteits primitive if and only if it is a class sum of its Brandt ideals. A set of other equivalent conditions on S is also obtained. Brandt semigroups are treated by Mcfadden and Schneider [4] and Preston and Clifford [6]. The reader is referred to Bruck [1] and Preston and Clifford [6] for many results and literature on these subjects. A semigroup is a non-null multiplicative associative system. Throughout S denotes a semigroup with zero. A subset A of S is a left (right) ideal of S if SA CA (AS CA). A is a two-sided ideal of S if it is both a left and a right ideal of S. A left ideal B of S is minimal if B # (0) and if it contains no proper left ideal of S. Similarly minimal right (two-sided) ideal is defined. S is simple if it contains no proper two-sided ideal of S and 52 / (0). The principal left (right) ideal generated by an element a in S is a U Sa (a UaS). S is said to be regular if for any a in S there exists x in S such that a = axa. e = ax and f = xa are idempotents and ea = a af. e (f) is called a left (right) unit of a. S is regular if and only if for any a in 5, a U aS = eS (a U SaSf) for some idempotent e (fi) in S. A regular semigroup with commuting idempotents is inverse. An idempotent e in S is primitive if for any idempotent f in S, ef M fe = f implies f0 or fe. S is completely simple if S is simple and contains a non-zero primitive idempotent. S is a Brandt semigroup if: