L∗-unipotent semigroups

El-Qallali, Abdulsalam

doi:10.1016/0022-4049(89)90018-2

Cited by 10 publications

(7 citation statements)

References 12 publications

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“…This is a special case of the subject discussed in the previous sections. The dual of £%*-unipotent semigroups was studied in [3] from which we conclude the following result. 6 for some x, y e S a , so that xae 6 =yae d .…”

Section: D2*-unipotent Semigroupsmentioning

confidence: 73%

Left regular bands of groups of left quotients

El-Qallali

1991

Glasgow Math. J.

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Introduction. In this paper we characterize semigroups 5 which have a semigroup Q of left quotients, where Q is an 9?-unipotent semigroup which is a band of groups. Recall that an 01-unipotent (or left inverse) semigroup S is one in which every 3?-class contains a unique idempotent. It is well-known that any 9?-unipotent semigroup 5 is a regular semigroup in which the set of idempotents is a left regular band in that efe = ef for any idempotents e, f in S. 3£-unipotent semigroups were studied by several authors, see for example [1] and [13]. Bailes [1] characterized 9?-unipotent semigroups which are bands of groups. This characterization extended the structure of inverse semigroups which are semilattices of groups. Recently, Gould studied in [7] the semigroup S which has a semigroup Q of left quotients where Q is an inverse semigroup which is a semilattice of groups.Many definitions of semigroups of quotients have been proposed and studied. For a survey, the reader may consult Weinert's paper [14]. These definitions have been motivated by corresponding definitions in ring theory. In this paper we are concerned with a concept of semigroups of left quotients adopted by Fountain and Petrich [5]. The definition proposed there is restricted to completely 0-simple semigroups of left quotients. The idea is that a completely 0-simple semigroup Q containing a subsemigroup S is a semigroup of left quotients of S if every element q in Q can be written as q = a~xb for some elements a, b in 5 with a 1 =t 0 and a" 1 the inverse of a in the group $?-class H a of Q. In this case 5 is also called a left order in Q. This definition and its dual were used in [5] to characterize semigroups 5 which have a completely 0-simple semigroup of quotients. An extension of this definition was used in [6] to obtain necessary and sufficient conditions for a semigroup 5 to have a bisimple inverse w-semigroup of left quotients. This extended definition was also used in [7] to characterize semigroups S which have a semigroup Q of left quotients where Q is an inverse semigroup which is a semilattice of groups. In this paper we consider the corresponding problem for 92-unipotent semigroups which are bands of groups.After preliminary results, we obtain in Section 2, the necessary and sufficient conditions for a semigroup 5 to have a semigroup Q of left quotients where Q is an !%-unipotent semigroup which is a band of groups. This result will be used in Section 3 together with the characterization of ^-unipotent semigroups which are bands of groups in terms of spined products to obtain an alternative structure for a semigroup 5 to have a left regular band of groups as a semigroup of left quotients. Section 4 is devoted to the case where the left orders are in a class of 9?*-unipotent semigroups.We use the notation and terminology of Howie [9]. Other undefined terms can be found in Fountain's paper [4].

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Section: D2*-unipotent Semigroupsmentioning

confidence: 73%

Left regular bands of groups of left quotients

El-Qallali

1991

Glasgow Math. J.

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“…Consequently, δ is a congruence on S. By Proposition 2.6 in ref. [20] and Lemma 2.7, the congruence δ has been already known to be the minimum adequate good congruence on S. Moreover, since the L * -inverse semigroup is always an IC abundant semigroup, by Lemma 2.4, S/δ is also an IC adequate semigroup. It turns out that S/δ is a type A semigroup.…”

Section: ⇐) This Part Is Easy Ifmentioning

confidence: 94%

The structure of ℒ*-inverse semigroups

Ren

Shum

2006

SCI CHINA SER A

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The concepts of L * -inverse semigroups and left wreath products of semigroups are introduced. It is shown that the L * -inverse semigroup can be described as the left wreath product of a type A semigroup Γ and a left regular band B together with a mapping which maps the semigroup Γ into the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the L * -inverse semigroups by using the left wreath products.

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“…The class of abundant semigroups encompasses a wide variety of semigroups, from regular semigroups and their full subsemigroups to bands of cancellative monoids ( [2]) and the multiplicative semigroups of PP rings. Abundant monoids first arose in connection with the theory of 5-systems.…”

Section: Introductionmentioning

confidence: 99%

“…Since abundant semigroups generalise regular semigroups, it is natural that, in the first instance, we should approach their structure theory by looking for generalisations of results from the theory of regular semigroups. The following papers contain some work along these lines: [1], [2], [3], [4], [5], [6], [9], [10], [11] and [17]. As we shall discuss below, this paper is one of a sequence in which we concentrate on the structure of a class of abundant semigroups whose natural partial order is compatible with the multiplication.…”

Section: Introductionmentioning

confidence: 99%

Rees matrix covers for a class of abundant semigroups

Lawson¹

1987

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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SynopsisRecently considerable attention has been paid to the study of locally inverse regular semigroups. McAlister [14] obtained a description of such semigroups as locally isomorphic images of regular Rees matrix semigroups over an inverse semigroup. The class of abundant semigroups originally arose from ‘homological’ considerations in the theory of S-systems: they are the semigroup theoretic analogue of PP-rings. Cancellative monoids, full subsemigroups of regular semigroups as well as the multiplicative semigroups of PP rings are abundant. The aim of this paper is to show how the structure theory described above for regular semigroups may be generalised to a class of abundant semigroups.

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L∗-unipotent semigroups

Cited by 10 publications

References 12 publications

Left regular bands of groups of left quotients

Left regular bands of groups of left quotients

The structure of ℒ*-inverse semigroups

Rees matrix covers for a class of abundant semigroups

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