Motivated by studying various classes of bisimple inverse semigroups with special semilattices of idempotents, and as a continuation of Reilly description of bisimple inverse ω − semigroups. In this paper, we introduce the concept of generalized Bruck–Reilly extensions. We call a (right) type B ω − semigroup with the generalized Bruck–Reilly extension a ∼ bisimple (right) type B ω − semigroup. After obtaining some basic properties and characterizations of this class of semigroups, we get sufficient and necessary conditions for an arbitrary ∼ bisimple (right) type B ω − semigroup to be a (right) type B semigroup. Finally, we get structure theorems for some special cases of such semigroups and give an application of our results.

On ∼ bisimple right type B ω− semigroups as model of some fractal structure

In this paper, some properties of good homomorphic images of abundant semigroups with a multiplicative quasi-adequate transversal are obtained. After that, good congruences on a class of abundant semigroups with such transversals are studied. We characterize the good congruences by using the so-called good congruence triple. Furthermore, we prove that any good congruence on a class of abundant semigroups with such transversals can be constructed by the good congruence triple.

Good congruences on abundant semigroups with multiplicative quasi-adequate transversals

A semigroup is called completely J(l)-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J(l)-simple semigroups form a quasivarity. Moreover, the construction of free completely J(l)-simple semigroups is given. It is found that a free completely J(l)-simple semigroup is just a free completely J*-simple semigroup and also a full subsemigroup of some completely simple semigroups.

Free completely J (ℓ)-simple Semigroups

In this article, we explore the multiplicative quasi-adequate transversals of an abundant semigroup. Let S be an abundant semigroup with multiplicative quasi-adequate transversals. Then the product of any two multiplicative quasi-adequate transversals is also a multiplicative quasi-adequate transversal. Moreover, all multiplicative quasi-adequate transversals of S form a rectangular band. Let S°and S ♭ be multiplicative quasi-adequate transversals of S, and let δ°(δ♭) be the δ-relation on S°(S ♭). Then there exists a bijection ϕ from S°/δ°onto S ♭/δ♭. In particular, if δ°and δ♭ are congruences, then the bijection ϕ is an isomorphism.

On the Multiplicative Quasi-Adequate Transversals of an Abundant Semigroup

In this paper, we study the class of ortho-u-monoids which are generalized orthogroups within the class of E(S)-semiabundant semigroups After introducing the concept of (∼)-Green’s relations, and obtaining some important properties of (∼)-Green’s relations and super E(S)-semiabundant semigroups, we have given the semilattice decomposition of ortho-u-monoids and a structure theorem for regular ortho-u-monoids The main techniques that we used in the study are the (∼)-Green’s relations, and the semi-spined product of semigroups

Ortho-u-monoids

The so-called split IC quasi-adequate semigroups are in the class of idempotent-connected quasi-adequate semigroups. It is proved that an IC quasi-adequate semigroup is split if and only if it has an adequate transversal. The structure of such semigroup whose band of idempotents is regular will be particularly investigated. Our obtained results enrich those results given by McAlister and Blyth on split orthodox semigroups.

Split quasi-adequate semigroups

The aim of this paper is to study a class of right pp semigroups, so-called pseudo-C-rpp semigroups. After obtaining some characterizations of pseudo-C-rpp semigroups, we establish a construction of such semigroups in terms of left normal bands and right C-rpp semigroups. In particular, a special case is considered.

Xiao Jiang Guo

Papers

Split quasi-adequate semigroups

Pseudo-C-rpp semigroups

Free completely J (ℓ)-simple Semigroups