Green’s relations and regularity in centralizers of permutations

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

A kernel-trace description of right congruences on an inverse semigroup is developed. It is shown that the trace mapping is a complete ∩ \cap homomorphism but not a ∨ \vee -homomorphism. However, the trace classes are intervals in the complete lattice of right congruences. In contrast, each kernel class has a maximum element, namely the principal right congruence on the kernel, but in general there is no minimum element in a kernel class. The kernel mapping preserves neither intersections nor joins. The set of axioms presented in [7] for right kernel systems is reviewed. A new set of axioms is obtained as a consequence of the fact that a right congruence is the intersection of the principal right congruences on the idempotent classes. Finally, it is shown that even though a congruence on a regular semigroup is the intersection of the principal congruences on the idempotent classes, the situation is not the same for right congruences on a regular semigroup. Right congruences on a regular, even orthodox, semigroup are not, in general, determined by their idempotent classes.

A regular semigroup S S is locally inverse if each local submonoid e S e eSe , e e an idempotent, is an inverse semigroup. It is shown that every locally inverse semigroup is an image of a regular Rees matrix semigroup, over an inverse semigroup, by a homomorphism θ \theta which is one-to-one on each local submonoid; such a homomorphism is called a local isomorphism. Regular semigroups which are locally isomorphic images of regular Rees matrix semigroups over semilattices are also characterized.

This article discusses some properties of regular semigroups. These properties are especially concerned with the relation of the regular semigroups to ideals, subsemigroups, groups, idempoten semigroups and invers semigroups. In addition, this paper also discusses the Cartesian product of two regular semigroups.
 
 Keywords:ideal, idempoten semigroup, inverse semigroup, regular semigroup, subsemigroup.

SynopsisA regular semigroup S is called V-regular if for any elements a, b ∈ S and any inverse (ab)′ of ab, there exists an inverse a′ of a and an inverse b′ of b such that (ab)′ = b′a′. A characterization of a V-regular semigroup is given in terms of its partial band of idempotents. The strongly V-regular semigroups form a subclass of the class of V-regular semigroups which may be characterized in terms of their biordered set of idempotents. It is shown that the class of strongly V-regular semigroups comprises the elementary rectangular bands of inverse semigroups (including the completely simple semigroups), a special class of orthodox semigroups (including the inverse semigroups), the strongly regular Baer semigroups (including the semigroups that are the multiplicative semigroup of a von Neumann regular ring), the full transformation semigroup on a set, and the semigroup of all partial transformations on a set.

A method for proving the embeddability of semigroup amalgams is introduced. After providing necessary and sufficient conditions in terms of representations for the weak embeddability of a semigroup amalgam, it successfully deals with the embedding of inverse semigroup amalgams into inverse semigroups and the embedding of an amalgam of regular semigroups whose core is full in each member.

In this survey we consider three kinds of algorithmic questions concerning varieties of semigroups. We are interested in identity problems, in the solvability of a system of equations and in the structure of all solutions of a given system. We study them in significant varieties of semigroups, monoids, groups, completely simple semigroups, completely regular semigroups (in particular semigroups satisfying the identity xr≈x) and involutary semigroups (in particular star regular semigroups and inverse semigroups).

Let C be a semiband with an inverse transversal \(C^{\circ}\). In [7], G.T. Song and F.L. Zhu construct a fundamental regular semigroup \(T_{C,C^{\circ}}\) with an inverse transversal \(T_{C,C^{\circ}}^{\circ}\). \(T_{C,C^{\circ}}\) is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation-although not the usual composition-is defined by means of composition. Any full regular subsemigroup T of \(T_{C,C^{\circ}}\) is a fundamental regular semigroup with inverse transversal \(T^{\circ}=T\cap T_{C,C^{\circ}}^{\circ}\). Moreover, any regular semigroup S with an inverse transversal \(S^{\circ}\) is proved to be an idempotent-separating coextension of a full regular subsemigroup T of some \(T_{C,C^{\circ}}\). By means of a full regular subsemigroup T of some \(T_{C,C^{\circ}}\) and by means of an inverse semigroup K satisfying some conditions, in this paper, we construct a regular semigroup \(T\ast K\) with inverse transversal \(T^{\circ}\ast K\) such that \(T^{\circ}\ast K\) is isomorphic to K and \((T\ast K)/\mu_{T\ast K}\) to T. Furthermore, it is proved that if S is a regular semigroup with an inverse transversal \(S^{\circ}\) then S can be constructed from the corresponding T and from \(S^{\circ}\) in this way.

Rees Matrix Covers and Semidirect Products of Regular Semigroups

SynopsisA Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters: (1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.

In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band ES of idempotents of S such that F∋e, ES∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of FS.

The minimal degree of an inverse semigroup S is the minimal cardinality of a set A such that S is isomorphic to an inverse semigroup of one-to-one partial transformations of A .The main result is a formula that expresses the minimal degree of a finite inverse semigroup S in terms of certain subgroups and the ordered structure of S .In fact, a representation of S by one-to-one partial transformations of the smallest possible set A is explicitly constructed in the proof of the formula.All known and some new results on the minimal degree follow as easy corollaries.

A construction of all coextensions of a regular semigroup $S$ by rectangular bands is obtained. The construction is analogous to Hall’s construction of orthodox semigroups as spined products of bands and inverse semigroups and reduces to that construction when $S$ is inverse. The results are specialized to provide a construction of the category of all normal coextensions of a regular semigroup.

This article aims to provide a comprehensive investigation into the concept of ideals within various semigroup classes, focusing on their properties, characterization, and role in algebraic structures. The research will cover classical semigroup theories, including regular semigroups, monoids, and groups, as well as less conventional semigroup classes, such as inverse semigroups, transformative semigroups, and idempotent semigroups. By delving into the intricacies of each semigroup type's ideal structures, this work seeks to expand the current understanding of these algebraic structures, offering new insights and potential applications in various fields, such as computer science, mathematics, and theoretical physics.