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.

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.

Completely Regular Semigroups with Generalized Strong Semilattice Decompositions

2005 ◽  
Vol 12 (02) ◽  
pp. 269-280 ◽  
Author(s):  
Xiangzhi Kong ◽  
K. P. Shum
Keyword(s):  
Sufficient Conditions ◽  
Normal Band ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Green’S Relation ◽  
Completely Regular Semigroups ◽  
Semilattice Of Semigroups ◽  
Strong Semilattice ◽  
Necessary And Sufficient ◽  
Characterization Theorems

The concept of ρG-strong semilattice of semigroups is introduced. By using this concept, we study Green's relation ℋ on a completely regular semigroup S. Necessary and sufficient conditions for S/ℋ to be a regular band or a right quasi-normal band are obtained. Important results of Petrich and Reilly on regular cryptic semigroups are generalized and enriched. In particular, characterization theorems of regular cryptogroups and normal cryptogroups are obtained.

Regular semigroups which are subdirect products of a band and a semilattice of groups

1973 ◽  
Vol 14 (1) ◽  
pp. 27-49 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Special Cases ◽  
Completely Regular Semigroups ◽  
Semilattice Of Semigroups ◽  
Subdirect Products ◽  
Completely Simple

In the study of the structure of regular semigroups, it is customary to impose several conditions restricting the behaviour of ideals, idempotents or elements. In a few instances, one may represent them as subdirect products of some much more restricted types of regular semigroups, e.g., completely (0-) simple semigroups, bands, semilattices, etc. In particular, studying the structure of completely regular semigroups, one quickly distinguishes certain special cases of interest when these semigroups are represented as semilattices of completely simple semigroups. In fact, this semilattice of semigroups may be built in a particular way, idempotents may form a subsemigroup, ℋ may be a congruence, and so on.

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 On the lattice of varieties of completely regular semigroups

1983 ◽  
Vol 35 (2) ◽  
pp. 227-235 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Variety Of Groups ◽  
Completely Regular Semigroups ◽  
Subdirect Products

AbstractSeveral morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).

Congruences on ideal nil-extension of completely regular semigroups

1998 ◽  
Vol 43 (5) ◽  
pp. 379-381
Author(s):  
Xueming Ren ◽  
Yuqi Guo ◽  
Jiaping Cen
Keyword(s):  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups ◽  
Nil Extension

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.

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