Quasi-adequate Semigroups Research Articles

Quasi-adequate semigroups with medial idempotents

The objective consists of two parts; one part is to investigate the structure of a right quasi-adequate semigroup [Formula: see text] containing a normal medial idempotent which is a left identity. Similarly, a left quasi-adequate semigroup case can be investigated. The other part is to combine the structure of both cases to establish a structure theorem for a quasi-adequate semigroup containing a normal medial idempotent.

Congruences on Split Quasi-Adequate Semigroups

In this paper, we give congruences on a split IC quasi-adequate semigroup by the congruence triple abstractly which consists of congruences on the structure component parts L, T, and R. We prove that the set of all congruences on this kind of semigroups is a complete lattice.

Adequate Transversals of Quasi-Adequate Semigroups

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands.

Split quasi-adequate semigroups

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.

Almost Factorizable IC Quasi-Adequate Semigroups

The aim of this article is to study a class of quasi-adequate semigroups, so-called almost factorizable idempotent-connected (IC) quasi-adequate semigroups. Our results enrich and extend the results of the second author on almost factorizable orthodox semigroups.

ABUNDANT SEMIGROUPS WITH QUASI-IDEAL S-ADEQUATE TRANSVERSALS

In this paper, the connection of the inverse transversal with the adequate transversal is explored. It is proved that if S is an abundant semigroup with an adequate transversal <TEX>$S^o$</TEX>, then S is regular if and only if <TEX>$S^o$</TEX> is an inverse semigroup. It is also shown that adequate transversals of a regular semigroup are just its inverse transversals. By means of a quasi-adequate semigroup and a right normal band, we construct an abundant semigroup containing a quasi-ideal S-adequate transversal and conversely, every such a semigroup can be constructed in this manner. It is simpler than the construction of Guo and Shum [9] through an SQ-system and the construction of El-Qallali [5] by W(E, S).

A new construction for abundant semigroups with multiplicative quasi-adequate transversals

In any abundant semigroup with a quasi-adequate transversal, we de ne two sets R and L and give some properties and characterizations associated with them. Then we give a structure theorem for abundant semigroups with multiplicative quasi-adequate transversals by means of two quasi-adequate semigroups R and L.

Good B-quasi-Ehresmann semigroups

A semigroup S with a sub-band B is called a good B-quasi-Ehresmann semigroup if it is a B-semiabundant semigroup satisfying the congruence condition such that aB(a*)B(b+)b+ ⊆ B((ab)+)abB((ab)*) for all a, b ∈ S. We show that every good B-quasi-Ehresmann semigroup has a global representation and a standard representation. As a special case, the structure of good quasi-adequate semigroups is described.

On q *-bisimple IC semigroups of type-E

A semigroup is called type-E if the band of its idempotents can be expressed as a direct product of a rectangular band and an ω-chain. For brevity, we call an IC *-bisimple quasi-adequate semigroup of type-E a q*-bisimple IC semigroup of type-E. In this paper, we characterize q*-bisimple semigroups by using some kind of generalized Bruck-Reilly extensions. As a consequence, some results concerning *-bisimple type-Aω-semigroups given by Asibong-Ibe (Semigroup Forum 31:99–117, 1985) are generalized.

A Class of Quasi-Adequate Semigroups

Let S be a quasi-adequate semigroup and the relation δ on S be defined by the rule: for a, b ∈ S, aδb if and only if E(a+)aE(a*) = E(b+)bE(b*) for some a+, a*, b+, b*. In this paper, we first construct an algebraic class called a QT-system and then prove that an abundant semigroup S is quasi-adequate on which δ is a congruence if and only if it is isomorphic to a QT-system.

Left semi-perfect abundant semi-groups of type W

We call a quasi-adequate semi-group whose set of idempotents forms a left [right] quasi-normal band a left [right] semi-perfect abundant semi-group. After obtaining some characterization theorems of such quasi-adequate semi-groups, we establish a structure for left [right] semi-perfect abundant semi-groups of type W. Our results generalize and strengthen the results of El-Qallali and Fountain on quasi-adequate semi-groups.

Left semi-perfect abundant semi-groups of type W

We call a quasi-adequate semi-group whose set of idempotents forms a left [right] quasi-normal band a left [right] semi-perfect abundant semi-group. After obtaining some characterization theorems of such quasi-adequate semi-groups, we establish a structure for left [right] semi-perfect abundant semi-groups of type W. Our results generalize and strengthen the results of El-Qallali and Fountain on quasi-adequate semi-groups.

Left Abundant Semigroups#

The aim of this paper is to study some special lpp-semigroups, namely, the left GC-lpp semigroups. After obtaining some properties and characterizations of such semigroups, we establish some structure theorems of this class of semigroups. In addition, we also consider some special cases. As an application, we describe the structure theorems of IC quasi-adequate semigroups whose idempotent band is a regular band.

On idempotent-connected quasi-adequate semigroups

The aim of this paper is to study a class of F-abundant semigroups, that is, u-IC quasi-adequate semigroups. After giving some characterizations of u-IC quasiadequate semigroups, we establish the structure of such semigroups in terms of the semidirect product of bands and cancellative monoids.

Quasi-adequate semigroups II

In this paper we describe the least fundamental adequate good congruence on an arbitrary typeW semigroupS as well as the largest superabundant full subsemigroup ofS and the largest full subsemigroup ofS which is a band of cancellative monoids. TypeW semigroups were introduced in [3] and a structure theorem for those semigroups was obtained. Here we introduce weak typeW semigroups by replacing the idempotent-connected property in typeW by one of its consequences. A structure theorem is obtained for such semigroups.

Quasi-adequate semigroups

SynopsisThe investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroup S we study the minimum good congruence γ such that S/γ is adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.