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Bicyclic semigroup

About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.


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TL;DR: In this article, a characterisation of all globally idempotent semigroups whose lattice of congruences is complemented is given, and an arbitrary semigroup has a complemented congruence lattice if and only if it is an inflation of a semigroup characterised in this way.
Abstract: Using the decomposition of a semigroup into itsI-classes, the paper gives a characterisation of all globally idempotent semigroups whose lattice of congruences is complemented. Furthermore, an arbitrary semigroup has a complemented congruence lattice if and only if it is an inflation of a semigroup characterised in this way. Thus the general problem of describing all semigroups with complemented congruence lattices is reduced to that of studying the question for the class of simple semigroups.

8 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that any almost unitary subsemigroup of a semigroup is closed in the containing semigroup and that the class of all left [right] regular semigroups is closed.
Abstract: We show that any almost unitary subsemigroup of a semigroup is closed in the containing semigroup and that the class of all left [right] regular semigroups is closed. Finally, we show that every globally idempotent ideal satisfying a seminormal permutation identity of a supersaturated semigroup is supersaturated.

8 citations

Journal ArticleDOI
TL;DR: In this article, the authors introduce the concepts of ordered quasi-ideals, ordered bi-ideal and ordered ternary semigroups and study the properties of these classes.
Abstract: . We introduce the concepts of ordered quasi-ideals, ordered bi-ideals in anordered ternary semigroup and study their properties. Also regular ordered ternarysemigroup is de ned and several ideal-theoretical characterizations of the regular orderedternary semigroups are furnished. 1. IntroductionThe literature of a ternary algebraic system was introduced by D. H. Lehmer[3] in 1932. He investigated certain ternary algebraic systems called triplexes whichturn out to be ternary groups. The notion of ternary semigroup was known toS.Banach. He showed by an example that ternary semigroup does not necessar-ily reduce to an ordinary semigroup. In [6] M. L. Santiago developed the theory ofternary semigroups. He focused his attention mainly to the study of regular ternarysemigroups, bi-ideals and ideals in ternary semigroups. The semigroup Z of all in-tegers under multiplication which plays a vital role in the literature of semigroup.The subset Z + of all positive integers of Z is a semigroup under multiplication.Now if we consider the subset Z of all negative integers of Z, then it is not a semi-group under multiplication. Taking these facts in mind D. H. Lehmer [3] introducedthe notion of ternary semigroup. Z is a natural example of a ternary semigroupunder the ternary multiplication. N. Kehayopulu in [5] developed the theory ofpo-semigroups. He mainly studied regular po-semigroups, ideals and bi-ideals inpo-semigroups. In 1999, Sang Keun Lee and Seong Gon Kang [4] gave charac-

8 citations

Journal ArticleDOI
TL;DR: It is shown how the existence of finite proper covers for semigroups in this quasivariety is a consequence of Ash’s powerful theorem for pointlike sets.
Abstract: The technique of covers is now well established in semigroup theory. The idea is, given a semigroup S, to find a semigroup Ŝ having a better understood structure than that of S, and an onto morphism θ of a specific kind from Ŝ to S. With the right conditions on θ, the behaviour of S is closely linked to that of Ŝ. If S is finite one aims to choose a finite Ŝ. The celebrated results for inverse semigroups of McAlister in the 1970’s form the flagship of this theory. Weakly left quasi-ample semigroups form a quasivariety (of algebras of type(2, 1)), properly containing the classes of groups, and of inverse, left ample, and weakly left ample semigroups. We show how the existence of finite proper covers for semigroups in this quasivariety is a consequence of Ash’s powerful theorem for pointlike sets. Our approach is to obtain a cover Ŝ of a weakly left quasi-ample semigroup S as a subalgebra of S × G, where G is a group. It follows immediately from the fact that weakly left quasi-ample semigroups form a quasivariety, that Ŝ is weakly left quasi-ample. We can then specialise our covering results to the quasivarieties of weakly left ample, and left ample semigroups. The latter have natural representations as (2, 1)-subalgebras of partial (one-one) transformations, where the unary operation takes a transformation α to the identity map in the domain of α. In the final part of this paper we consider representations of weakly left quasi-ample semigroups.

8 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202312
202229
20217
20203
20194
201810