Quasi-adequate semigroups

El-Qallali, Abdulsalam; Fountain, John

doi:10.1017/s0308210500012658

Cited by 62 publications

(64 citation statements)

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“…In 1981, El-Qallali and Fountain [7] generalised this result to abundant semigroups with band of idempotents, and satisfying the idempotent connected condition (IC) [8]. They described such a semigroup S having a band of idempotents B as a spined product of W B and S/δ B , where δ B is the analogue of the least inverse Ehresmann-Schein-Nambooripad (ESN) Theorem, and its many extensions due to Armstrong [1,2], Lawson [32], Meakin [35,36] and Nambooripad [38][39][40].…”

Section: Prefacementioning

confidence: 99%

“…we have quasi-adequate semigroups [7]. A quasi-adequate semigroup is an abundant semigroup whose set of idempotents forms a subsemigroup.…”

Section: Abundant Semigroupsmentioning

confidence: 99%

“…We wish to find a closed form for γ B . These ideas have been investigated in [6], [7], [14] and [46]. Let S be a weakly B-orthodox semigroup with (WIC) (resp.…”

Section: An Analogue Of the Least Inverse Congruencementioning

confidence: 99%

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Beyond regular semigroups

Wang

2015

Semigroup Forum

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The topic of this thesis is the class of weakly U-abundant semigroups. This class is very wide, containing inverse, orthodox, regular, ample, adequate, quasiadequate, concordant, abundant, restriction, Ehresmann and weakly abundant semigroups. A semigroup S with subset of idempotents U is weakly U-abundant if every R U -class and every L U -class contains an idempotent of U, where R U and L U are relations extending the well known Green's relations R and L. We assume throughout that our semigroups satisfy a condition known as the Congruence Condition (C).We take several approaches to weakly U-abundant semigroups. Our first results describe those that are analogous to completely simple semigroups. Together with an existing result of Ren this determines the structure of those weakly Uabundant semigroups that are analogues of completely regular semigroups, that is, they are superabundant. Our description is in terms of a semilattice of rectangular bands of monoids.The second strand is to aim for an extension of the Hall-Yamada theorem for orthodox semigroups as spined products of inverse semigroups and fundamental orthodox semigroups. To this end we consider weakly B-orthodox semigroups, where B is a band. We note that if B is a semilattice then a weakly B-orthodox semigroup is exactly an Ehresmann semigroup. We provide a description of a weakly B-orthodox semigroup S as a spined product of a fundamental weakly Borthodox semigroup S B (depending only on B) and S/γ B , where B is isomorphic to B and γ B is the analogue of the least inverse congruence on an orthodox semigroup. This result is an analogue of the Hall-Yamada theorem for orthodox semigroups. In the case that B is a normal band, or S is weakly B-superabundant, we find a closed form δ B for γ B , which simplifies our result to a straightforward form.i For the above to work smoothly in the case S is weakly B-superabundant, we need to find a canonical fundamental weakly B-superabundant subsemigroup of S B . This we do, and give the corresponding answers in the case of the Hall semigroup W B and a number of intervening semigroups.We then change our direction. A celebrated result of Nambooripad shows that regular semigroups are determined by ordered groupoids built over a regular biordered set. Our aim, achieved at the end of the thesis, is to extend Nambooripad's work to weakly U-regular semigroups, that is, weakly U-abundant semigroups with (C) and U generating a regular subsemigroup whose set of idempotents is U.As an intervening step we consider weakly B-orthodox semigroups in this light. We take two approaches. The first is via a new construction of an inductive generalised category over a band. In doing so we produce a new approach to characterising orthodox semigroups, by using inductive generalised groupoids.We show that the category of weakly B-orthodox semigroups is isomorphic to the category of inductive generalised categories over bands. Our approach is influenced by that of Nambooripad, however, there are significant differences in strateg...

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Section: Prefacementioning

confidence: 99%

“…we have quasi-adequate semigroups [7]. A quasi-adequate semigroup is an abundant semigroup whose set of idempotents forms a subsemigroup.…”

Section: Abundant Semigroupsmentioning

confidence: 99%

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Beyond regular semigroups

Wang

2015

Semigroup Forum

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“…At first, we recall some basic facts about the relation [3] if and only if each * L class and each * R class contains at least one idempotent. An abundant semigroup S is called quasi-adequate [2] if its set of idempotents constitutes a subsemigroup ( i.e., its set of idempotents is a band). Moreover, a quasi-adequate semigroup is called adequate [6] if its bands of idempotents is a semilattice (i.e., the idempotents commute).…”

Section: Preliminariesmentioning

confidence: 99%

Some Notes on Abundant Quasi-Ideals of Ample Semigroups

Li¹

2015

Proceedings of the International Conference on Advances in Energy, Environment and Chemical Engineering

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a chunhuali66@163.com Keywords: natural partial orders; abundant quasi-ideals; (bi-)order ideals; ample semigroups. Abstract. The aim of this paper is to study ample semigroups by using natural partial orders on abundant semigroups. After giving some properties and characterizations of abundant quasi-ideals on ample semigroups, we prove that there exists an idempotent-separating good congruence  on an abundant quasi-ideal M of an arbitrary ample semigroup S, which can be extended to an idempotent-separating good congruence on S. In this paper, we develop an approach to the structure of ample semigroups inspired by the semigroup algebra theory of Mario petrich for inverse semigroups.

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“…In [3] the authors initiate the study of quasi-adequate semigroups and provide a structure theorem based on spined products. A semigroup is said to be quasi-adequate if it is abundant and its idempotents form a subsemigroup.…”

Section: Proofmentioning

confidence: 99%