Good congruences on abundant semigroups with quasi-ideal adequate transversals

2019 ◽  
Vol 56 (3) ◽  
pp. 323-334
Author(s):  
Aifa Wang
Keyword(s):  
Complete Lattice ◽  
Abundant Semigroup ◽  
Structure Component ◽  
Abundant Semigroups ◽  
Adequate Transversal

Abstract The aim of this paper is to study the congruences on abundant semigroups with quasi-ideal adequate transversals. The good congruences on an abundant semigroup with a quasi-ideal adequate transversal S° are described by the equivalence triple abstractly which consists of equivalences on the structure component parts I, S° and Λ. Also, it is shown that the set of all good congruences on this kind of semigroup forms a complete lattice.

scholarly journals The characterization and the product of quasi-Ehresmann transversals

Filomat ◽  
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.

Some Properties Associated with Adequate Transversals

2011 ◽  
Vol 54 (3) ◽  
pp. 487-497 ◽  
Author(s):  
Xiangjun Kong
Keyword(s):  
Regularity Condition ◽  
Inverse Transversal ◽  
Abundant Semigroup ◽  
Classic Result ◽  
Equivalent Conditions ◽  
Adequate Transversal

AbstractIn this paper, another relationship between the quasi-ideal adequate transversals of an abundant semigroup is given. We introduce the concept of a weakly multiplicative adequate transversal and the classic result that an adequate transversal is multiplicative if and only if it is weakly multiplicative and a quasi-ideal is obtained. Also, we give two equivalent conditions for an adequate transversal to be weakly multiplicative. We then consider the case when I and Λ (defined below) are bands. This is analogous to the inverse transversal if the regularity condition is adjoined.

scholarly journals On refined generalised quasi-adequate transversals

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 299-313
Author(s):  
Xiangjun Kong ◽  
Pei Wang
Keyword(s):  
Structure Theorem ◽  
Interesting Property ◽  
Product Structure ◽  
Abundant Semigroup ◽  
Property A ◽  
The Real ◽  
Spined Product ◽  
Abundant Semigroups

Some properties and characterizations for abundant semigroups with generalised quasiadequate transversals are explored. In such semigroups, an interesting property [?a,b ? Re1S, VSo(a) ? VSo (b) ? 0 ? VSo (a) = VSo (b)] is investigated and thus the concept of refined generalised quasi-adequate transversals, for short, RGQA transversals is introduced. It is shown that RGQA transversals are the real common generalisations of both orthodox transversals and adequate transversals in the abundant case. Finally, by means of two abundant semigroups R and L, a spined product structure theorem for an abundant semigroup with a quasi-ideal RGQA transversal is established.

scholarly journals The product of quasi-ideal refined generalised quasi-adequate transversals

2019 ◽  
Vol 17 (1) ◽  
pp. 43-51
Author(s):  
Xiangjun Kong ◽  
Pei Wang ◽  
Yonghong Wu
Keyword(s):  
Regularity Condition ◽  
Rectangular Band ◽  
Abundant Semigroup ◽  
The Real ◽  
Adequate Transversal

Abstract As the real common generalisations of both orthodox transversals and adequate transversals in the abundant case, the concept of refined generalised quasi-adequate transversal, for short, RGQA transversal was introduced by Kong and Wang. In this paper, an interesting characterization for a generalised quasi-adequate transversal to be refined is acquired. It is shown that the product of every two quasi-ideal RGQA transversals of the abundant semigroup S satisfying the regularity condition is a quasi-ideal RGQA transversal of S and that all quasi-ideal RGQA transversals of S compose a rectangular band. The related results concerning adequate transversals are generalised and enriched.

scholarly journals Self-injectivity of semigroup algebras

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.

scholarly journals On abundant semigroups withweak normal idempotents

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1241-1249
Author(s):  
Haizhou Chao ◽  
Xiangfei Ni
Keyword(s):  
Abundant Semigroup ◽  
Abundant Semigroups ◽  
Proper Subclass

A weak normal idempotent of an abundant semigroup was introduced by Guo [7]. In this paper, weak normal idempotents and normal idempotents of abundant semigroups are respectively characterized in many different ways. These results enable us to obtain an example which shows that the class of normal idempotents of abundant semigroups is a proper subclass of normal idempotents of abundant semigroups. Furthermore, this example tell us that there exists a non-regular abundant semigroup containing a weak normal idempotent. At last, we investigate the relationships between weak normal idempotents and normal idempotents and deduce that the main result of [2] can not be generalized into the class of abundant semigroups.

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

2008 ◽  
Vol 15 (04) ◽  
pp. 653-666 ◽  
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.

REES MATRIX COVERS FOR TIGHT ABUNDANT SEMIGROUPS

2010 ◽  
Vol 03 (03) ◽  
pp. 409-425
Author(s):  
Xiaojiang Guo ◽  
K. P. Shum ◽  
Yongqian Zhu
Keyword(s):  
Rees Matrix Semigroup ◽  
Abundant Semigroup ◽  
Regular Semigroups ◽  
Regular Elements ◽  
Abundant Semigroups ◽  
Matrix Semigroup

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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