REES MATRIX COVERS FOR TIGHT ABUNDANT SEMIGROUPS

Rees matrix covers for regular semigroups were first studied by McAlister in 1984. Lawson extended McAlister's results to abundant semigroups in 1987. We consider here a semigroup whose set of regular elements forms a subsemigroup, named tight semigroups. In this paper, it is proved that an abundant semigroup is tight and locally E-solid if and only if it is an F-local isomorphic image of an abundant Rees matrix semigroup [Formula: see text] over a tight E-solid abundant semigroup T, where the entries of the sandwich matrix P of [Formula: see text] are regular elements of T. Our results enrich the result of Lawson on Rees matrix covers for a class of abundant semigroups and extend the results of McAlister on Rees matrix covers for regular semigroups.

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The characterization and the product of quasi-Ehresmann transversals

Filomat ◽

10.2298/fil1907051k ◽

2019 ◽

Vol 33 (7) ◽

pp. 2051-2060

Author(s):

Xiangjun Kong ◽

Pei Wang

Keyword(s):

Abundant Semigroup ◽

Regular Semigroups ◽

The Real ◽

Abundant Semigroups

Wang (Filomat 29(5), 985-1005, 2015) introduced and investigated quasi-Ehresmann transversals of semi-abundant semigroups satisfy conditions (CR) and (CL) as the generalizations of orthodox transversals of regular semigroups in the semi-abundant case. In this paper, we give two characterizations for a generalized quasi-Ehresmann transversal to be a quasi-Ehresmann transversal. These results further demonstrate that quasi-Ehresmann transversals are the ?real? generalizations of orthodox transversals in the semi-abundant case. Moreover, we obtain the main result that the product of any two quasi-ideal quasi-Ehresmann transversals of a semi-abundant semigroup S which satisfy the certain conditions is a quasi-ideal quasi-Ehresmann transversal of S.

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FAITHFUL FUNCTORS FROM CANCELLATIVE CATEGORIES TO CANCELLATIVE MONOIDS WITH AN APPLICATION TO ABUNDANT SEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196705002451 ◽

2005 ◽

Vol 15 (04) ◽

pp. 683-698 ◽

Cited By ~ 9

Author(s):

VICTORIA GOULD ◽

MARK KAMBITES

Keyword(s):

Inverse Semigroup ◽

Self-injectivity of semigroup algebras

Open Mathematics ◽

10.1515/math-2020-0023 ◽

2020 ◽

Vol 18 (1) ◽

pp. 333-352

Author(s):

Junying Guo ◽

Xiaojiang Guo

Keyword(s):

Regular Semigroup ◽

Necessary Conditions ◽

Semigroup Algebra ◽

Semigroup Algebras ◽

Abundant Semigroup ◽

Regular Semigroups ◽

Abundant Semigroups ◽

Sufficient And Necessary Conditions ◽

Finite Regular Semigroup

Abstract It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K 0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S] is a right (respectively, left) self-injective imply that S is finite? (Semigroup Algebras, Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K 0[S] being right (left) self-injective when K 0[S] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.

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A Structure Theorem of Regular ${\cal H}^\sharp$-Cryptographs

Algebra Colloquium ◽

10.1142/s100538670800062x ◽

2008 ◽

Vol 15 (04) ◽

pp. 653-666 ◽

Cited By ~ 3

Author(s):

Xiangzhi Kong ◽

Zhiling Yuan ◽

K. P. Shum

Keyword(s):

Structure Theorem ◽

Abundant Semigroup ◽

Regular Semigroups ◽

Completely Regular ◽

Abundant Semigroups ◽

Completely Regular Semigroups ◽

Semilattice Of Semigroups ◽

Strong Semilattice ◽

Green Relations ◽

Strong Semilattice Of Semigroups

A new set of generalized Green relations is given in studying the [Formula: see text]-abundant semigroups. By using the generalized strong semilattice of semigroups recently developed by the authors, we show that an [Formula: see text]-abundant semigroup is a regular [Formula: see text]-cryptograph if and only if it is an [Formula: see text]-strong semilattice of completely [Formula: see text]-simple semigroups. This result not only extends the well known result of Petrich and Reilly from the class of completely regular semigroups to the class of semiabundant semigroups, but also generalizes a well known result of Fountain on superabundant semigroups from the class of abundant semigroups to the class of semiabundant semigroups.

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Semiprimeness of semigroup algebras

Open Mathematics ◽

10.1515/math-2021-0026 ◽

2021 ◽

Vol 19 (1) ◽

pp. 803-832

Author(s):

Junying Guo ◽

Xiaojiang Guo

Keyword(s):

Regular Semigroup ◽

Regularity Condition ◽

Necessary Conditions ◽

Semigroup Algebra ◽

Semigroup Algebras ◽

Abundant Semigroup ◽

Locally Finite ◽

Regular Semigroups ◽

Abundant Semigroups ◽

Sufficient And Necessary Conditions

Abstract Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and study D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices of a semigroup. Based on D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebra A {\mathcal{A}} with unity, A {\mathcal{A}} is primitive (prime) if and only if so is M n ( A ) {M}_{n}\left({\mathcal{A}}) . Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.

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Rees matrix semigroups over inverse semigroups

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500014499 ◽

1986 ◽

Vol 102 (1-2) ◽

pp. 61-90 ◽

Cited By ~ 5

Author(s):

F. J. Pastijn ◽

Mario Petrich

Keyword(s):

Inverse Semigroup ◽

Homomorphic Image ◽

Inverse Semigroups ◽

Rees Matrix Semigroup ◽

Regular Semigroups ◽

Rees Matrix Semigroups ◽

Special Cases ◽

Matrix Semigroup ◽

Matrix Semigroups ◽

Completely Simple

SynopsisA Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

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On Uσ-Abundant Semigroups

Algebra Colloquium ◽

10.1142/s100538671200003x ◽

2012 ◽

Vol 19 (01) ◽

pp. 41-52 ◽

Cited By ~ 5

Author(s):

Xueming Ren ◽

Qingyan Yin ◽

K. P. Shum

Keyword(s):

Structure Theorem ◽

Abundant Semigroup ◽

Regular Semigroups ◽

Permutation Identity ◽

Spined Product ◽

Abundant Semigroups ◽

The Relationship ◽

Ehresmann Semigroups

A U-abundant semigroup whose subset U satisfies a permutation identity is said to be Uσ-abundant. In this paper, we consider the minimum Ehresmann congruence δ on a Uσ-abundant semigroup and explore the relationship between the category of Uσ-abundant semigroups (S,U) and the category of Ehresmann semigroups (S,U)/δ. We also establish a structure theorem of Uσ-abundant semigroups by using the concept of quasi-spined product of semigroups. This generalizes a result of Yamada for regular semigroups in 1967 and a result of Guo for abundant semigroups in 1997.

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Semi-abundant semigroups with quasi-Ehresmann transversals

Filomat ◽

10.2298/fil1505985w ◽

2015 ◽

Vol 29 (5) ◽

pp. 985-1005

Author(s):

Shoufeng Wang

Keyword(s):

Structure Theorem ◽

Regular Semigroups ◽

Abundant Semigroups

Chen (Communications in Algebra 27(2), 4275-4288, 1999) introduced and investigated orthodox transversals of regular semigroups. In this paper, we initiate the investigation of quasi-Ehresmann transversals of semi-abundant semigroups which are generalizations of orthodox transversals of regular semigroups. Some interesting properties associated with quasi-Ehresmann transversals are established. Moreover, a structure theorem of semi-abundant semigroups with generalized bi-ideal strong quasi-Ehresmann transversals is obtained. Our results generalize and enrich Chen?s results.

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Generators and relations of Rees matrix semigroups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500020472 ◽

1999 ◽

Vol 42 (3) ◽

pp. 481-495 ◽

Cited By ~ 16

Author(s):

H. Ayik ◽

N. Ruškuc

Keyword(s):

Rees Matrix Semigroup ◽

Finitely Generated ◽

Finite Generation ◽

Generators And Relations ◽

Rees Matrix Semigroups ◽

Finite Presentability ◽

Matrix Semigroup ◽

Finitely Presented ◽

Matrix Semigroups ◽

The Ideal

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

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Left and right regular elements of the semigroups of transformations preserving an equivalence relation and a cross-section

Asian-European Journal of Mathematics ◽

10.1142/s179355711950058x ◽

2019 ◽

Vol 12 (04) ◽

pp. 1950058

Author(s):

Nares Sawatraksa ◽

Chaiwat Namnak ◽

Ronnason Chinram

Keyword(s):

Cross Section ◽

Equivalence Relation ◽

Regular Semigroups ◽

Completely Regular ◽

Regular Elements ◽

Left Regular ◽

Completely Regular Semigroups ◽

Cross Section Formula ◽

Left And Right

Let [Formula: see text] be the semigroup of all transformations on a set [Formula: see text]. For an arbitrary equivalence relation [Formula: see text] on [Formula: see text] and a cross-section [Formula: see text] of the partition [Formula: see text] induced by [Formula: see text], let [Formula: see text] [Formula: see text] Then [Formula: see text] and [Formula: see text] are subsemigroups of [Formula: see text]. In this paper, we characterize left regular, right regular and completely regular elements of [Formula: see text] and [Formula: see text]. We also investigate conditions for which of these semigroups to be left regular, right regular and completely regular semigroups.

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