Amalgamation and inverse and regular semigroups

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.

GENERALIZED INVERSE SEMIGROUPS AND AMALGAMATION

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.

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).

For any permutation σ on N = {1,2,…,n}: (i) Green’s relations are characterized in the centralizer C(σ) of σ (relative to the semigroup PTn of partial transformations on N); and (ii) a criterion is given for C(σ) to be a regular semigroup (inverse semigroup, union of groups).

In this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ:S→Tbetween regular semigroups. This is a mapping such that (ab)θ ≦ aθ bθin the natural partial order onT.

We study inverse semigroup amalgams [S 1,S 2;U], where S 1 and S 2 are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S 1 and S 2. We use a modified version of Bennett’s work on the structure of Schützenberger graphs of the ℛ-classes of S 1* U S 2 to state sufficient conditions for the amalgamated free products S 1* U S 2 having decidable word problem.

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 finite generation and presentation of Schützenberger products of semigroups are investigated. A general necessary and sufficient condition is established for finite generation. The Schützenberger product of two groups is finitely presented as an inverse semigroup if and only if the groups are finitely presented, but is not finitely presented as a semigroup unless both groups are finite.