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.

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.

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.

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.

Abundant semigroups with a *-normal idempotent

2017 ◽  
Vol 67 (4) ◽  
Author(s):  
Yonglin Hou ◽  
Junying Guo ◽  
Xiaojiang Guo
Keyword(s):  
Structure Theorem ◽  
Abundant Semigroups

AbstractThe notion of *-normal idempotents is introduced. The structure theorem for abundant semigroups with a *-normal idempotent is obtained. As its applications, we establish the construction theorem of naturally ordered abundant semigroups with a greatest idempotent.

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.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

On Lattice-Ordered Rees Matrix Γ-Semigroups

2013 ◽  
Vol 59 (1) ◽  
pp. 209-218 ◽  
Author(s):  
Kostaq Hila ◽  
Edmond Pisha
Keyword(s):  
Structure Theorem ◽  
Lattice Of Congruences

Abstract The purpose of this paper is to introduce and give some properties of l-Rees matrix Γ-semigroups. Generalizing the results given by Guowei and Ping, concerning the congruences and lattice of congruences on regular Rees matrix Γ-semigroups, the structure theorem of l-congruences lattice of l - Γ-semigroup M = μº(G : I; L; Γe) is given, from which it follows that this l-congruences lattice is distributive.

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