scholarly journals Semi-abundant semigroups with quasi-Ehresmann transversals

Filomat ◽  
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.

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.

Weakly U-abundant semigroups with strong Ehresmann transversals

2018 ◽  
Vol 68 (3) ◽  
pp. 549-562
Author(s):  
Dandan Yang
Keyword(s):  
Structure Theorem ◽  
Direct Application ◽  
Regular Semigroups ◽  
Abundant Semigroups

Abstract The class of weakly U-abundant semigroups is an important source of non-regular semigroups, and it is well studied by semigroup theorists in recent years. An important subclass, called Ehresmann monoids, is deeply investigated by Branco, Gomes and Gould in 2014. In this paper, we are concerned with weakly U-abundant semigroups with strong Ehresmann transversals. Our aim is to give a structure theorem for such semigroups following the standard “Rees Theorem” type approach. As a direct application of the main result, we reobtain the structure theorem of abundant semigroups with quasi ideal adequate transversals by Chen in 2000.

A STRUCTURE THEOREM FOR PERFECT ABUNDANT SEMIGROUPS

2008 ◽  
Vol 01 (01) ◽  
pp. 69-76 ◽  
Author(s):  
Xiaojiang Guo ◽  
K. P. Shum
Keyword(s):  
Direct Product ◽  
Rectangular Band ◽  
Structure Theorem ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Cancellative Monoid ◽  
Abundant Semigroups ◽  
Completely Regular Semigroups ◽  
Strong Semilattice

The direct product of a cancellative monoid and a rectangular band is called a can-cellative plank. In this paper, we describe the semigroups which can be expressed as a strong semilattice of cancellative planks. Our result not only generalizes the well known 1951 Clifford theorem for completely regular semigroups having central idempotents, but also the theorem for C-rpp monoids, that is, left abundant monoids having central idempotents, given by Fountain in 1977. Some recent results of the authors concerning rpp semigroups belonging to a class we call perfect are strengthened.

On Uσ-Abundant Semigroups

2012 ◽  
Vol 19 (01) ◽  
pp. 41-52 ◽  
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.

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.

On a class of Dubreil-Jacotin regular semigroups and a construction of Yamada

1977 ◽  
Vol 77 (1-2) ◽  
pp. 145-150 ◽  
Author(s):  
T. S. Blyth
Keyword(s):  
Structure Theorem ◽  
Present Approach ◽  
Inverse Semigroups ◽  
Regular Semigroups ◽  
Structure Theorems

SynopsisIn the publication [2] we obtained some structure theorems for certain Dubreil-Jacotin regular semigroups. A crucial observation in the course of investigating these types of ordered regular semigroups was that the (ordered) band of idempotents was normal. This is characteristic of a class of semigroups studied by Yamada [5] and called generalised inverse semigroups. Here we specialise a construction of Yamada to obtain a structure theorem that complements those in [2], The important feature of the present approach is the part played by the greatest elements that exist in each of the components in the semilattice decompositions involved.

scholarly journals On Naturally Ordered Abundant Semigroups with an Adequate Monoid Transversal

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Wei Chen
Keyword(s):  
Structure Theorem ◽  
Abundant Semigroups

In this paper, we study a class of naturally ordered abundant semigroups with an adequate monoid transversal, namely, naturally ordered concordant semigroups with an adequate monoid transversal. After giving some properties of such semigroups, we obtain a structure theorem for naturally ordered concordant semigroups with an adequate monoid transversal.

KNITTED SEMILATTIC STRUCTURE OF REGULAR ${\cal H}^\dag$-CYBER GROUPS

2011 ◽  
Vol 04 (03) ◽  
pp. 545-557
Author(s):  
Yu Su ◽  
Xiangzhi Kong
Keyword(s):  
Recent Result ◽  
Classical Theorem ◽  
Regular Semigroups ◽  
Abundant Semigroups ◽  
Structure Theorems ◽  
Semilattice Of Semigroups ◽  
Normal Formula

We consider regular [Formula: see text]-cyber groups in the class of [Formula: see text]-abundant semigroups. By using knitted semilattice of semigroups, we give some structure theorems for regular [Formula: see text]-cyber groups, right quasi-normal [Formula: see text]-cyber groups and normal [Formula: see text]-cyber groups. Our main result generalizes a classical theorem of Petrich- Reilly on normal cryptic groups from the class of regular semigroups to the class of generalized abundant semigroups and also entriches a recent result of Guo-Shum on left cyber groups.

scholarly journals A NEW CONSTRUCTION FOR REGULAR SEMIGROUPS WITH QUASI-IDEAL ORTHODOX TRANSVERSALS

2009 ◽  
Vol 86 (2) ◽  
pp. 177-187 ◽  
Author(s):  
XIANGJUN KONG ◽  
XIANZHONG ZHAO
Keyword(s):  
Regular Semigroup ◽  
Structure Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
New Construction ◽  
Necessary And Sufficient ◽  
Equivalent Conditions ◽  
Orthodox Semigroups

AbstractIn any regular semigroup with an orthodox transversal, we define two sets R and L using Green’s relations and give necessary and sufficient conditions for them to be subsemigroups. By using R and L, some equivalent conditions for an orthodox transversal to be a quasi-ideal are obtained. Finally, we give a structure theorem for regular semigroups with quasi-ideal orthodox transversals by two orthodox semigroups R and L.

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.

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