scholarly journals Idempotent-generated regular semigroups

1973 ◽  
Vol 15 (1) ◽  
pp. 27-34 ◽  
Author(s):  
C. Eberhart ◽  
W. Williams ◽  
L. Kinch
Keyword(s):  
Regular Semigroup ◽  
Regular Semigroups

Suppose S is a regular semigroup and E is its set of idempotents. If E is subsemigroup of S, then S has been called orthodox and studied recently by Hall [3], Meakin [6], and Yamada [8]. In this paper we assume that E is not (necessarily) a subsemigroup of S and consider the subsemigroup generated by E, denoted <E>. If E denotes the set of all elements of S which can be written E, denoted <E>. If E denotes the set of all elements of S which can be written as the product of n (not necessarily distinct) idempotents of S, then . We show that <E> is always a regular subsemigroup of S and investigate relationships between it and S. The case where <E> = S is of particular interest to us; such semigroups will be referred to as idempotent-generated regular semi- groups.

scholarly journals Regular semigroups, fundamental semigroups and groups

1980 ◽  
Vol 29 (4) ◽  
pp. 475-503 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Direct Product ◽  
Homomorphic Image ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Fundamental Semigroups

AbstractIn this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

scholarly journals Enlargements of regular semigroups

1996 ◽  
Vol 39 (3) ◽  
pp. 425-460 ◽  
Author(s):  
M. V. Lawson
Keyword(s):  
Regular Semigroup ◽  
Semigroup Theory ◽  
Regular Semigroups

We introduce a class of regular extensions of regular semigroups, called enlargements; a regular semigroup T is said to be an enlargement of a regular subsemigroup S if S = STS and T = TST. We show that S and T have many properties in common, and that enlargements may be used to analyse a number of questions in regular semigroup theory.

scholarly journals A special sublattice of the congruence lattice of a regular semigroup

1997 ◽  
Vol 40 (3) ◽  
pp. 457-472 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Distributive Lattice ◽  
Regular Semigroup ◽  
Congruence Lattice ◽  
Regular Semigroups ◽  
Generators And Relations ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Special Cases ◽  
Completely Regular Semigroups ◽  
Completely Simple

Let S be a regular semigroup and be its congruence lattice. For ρ ∈ , we consider the sublattice Lρ of generated by the congruences pw where w ∈ {K, k, T, t}* and w has no subword of the form KT, TK, kt, tk. Here K, k, T, t are the operators on induced by the kernel and the trace relations on . We find explicitly the least lattice L whose homomorphic image is Lρ for all ρ ∈ and represent it as a distributive lattice in terms of generators and relations. We also consider special cases: bands of groups, E-unitary regular semigroups, completely simple semigroups, rectangular groups as well as varieties of completely regular semigroups.

REGULAR SEMIGROUPS WITH NORMAL IDEMPOTENTS

2017 ◽  
Vol 103 (1) ◽  
pp. 116-125
Author(s):  
XIANGFEI NI ◽  
HAIZHOU CHAO
Keyword(s):  
Regular Semigroup ◽  
Simple Method ◽  
Regular Semigroups

In this paper, we investigate regular semigroups that possess a normal idempotent. First, we construct a nonorthodox nonidempotent-generated regular semigroup which has a normal idempotent. Furthermore, normal idempotents are described in several different ways and their properties are discussed. These results enable us to provide conditions under which a regular semigroup having a normal idempotent must be orthodox. Finally, we obtain a simple method for constructing all regular semigroups that contain a normal idempotent.

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 inverses of products of idempotents in regular semigroups

1972 ◽  
Vol 13 (3) ◽  
pp. 335-337 ◽  
Author(s):  
D. G. Fitz-Gerald
Keyword(s):  
Regular Semigroup ◽  
Regular Semigroups

Let E be the set of idempotents of a regular semigroup; we prove that V(En) = En+1 (see below for the meaning of this notation). This generalizes a result of Miller and Clifford ([3], theorem 4, quoted as exercise 3(b), p. 61, of Clifford and Preston [1]) and the converse, proved by Howie and Lallement ([2], lemma 1.1), which together establish the case n = 1. As a corollary, we deduce that the subsemigroup generated by the idempotents of a regular semigroup is itself regular.

scholarly journals Eventually Pointed Principally Ordered Regular Semigroups

2020 ◽  
Vol 24 (2) ◽  
pp. 139
Author(s):  
G.A. Pinto
Keyword(s):  
Positive Integer ◽  
Regular Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Completely Simple

An ordered regular semigroup, , is said to be principally ordered if for every  there exists . A principally ordered regular semigroup is pointed if for every element,  we have . Here we investigate those principally ordered regular semigroups that are eventually pointed in the sense that for all  there exists a positive integer, , such that . Necessary and sufficient conditions for an eventually pointed principally ordered regular semigroup to be naturally ordered and to be completely simple are obtained. We describe the subalgebra of  generated by a pair of comparable idempotents  and such that . 

scholarly journals Group congruences on eventually regular semigroups

1988 ◽  
Vol 45 (3) ◽  
pp. 320-325 ◽  
Author(s):  
S. Hanumantha Rao ◽  
P. Lakshmi
Keyword(s):  
Regular Semigroup ◽  
Group Congruence ◽  
Regular Semigroups ◽  
Eventually Regular Semigroup ◽  
Lattice Of Congruences

AbstractA characterization of group congruences on an eventually regular semigroup S is provided. It is shown that a group congruence is dually right modular in the lattice of congruences on S. Also for any group congruence ℸ and any congruence p on S, ℸ Vp and kernel ℸ Vp are described.

Regular Semigroups Which Are Amalgamation Bases for Finite Semigroups

2007 ◽  
Vol 14 (02) ◽  
pp. 245-254 ◽  
Author(s):  
Kunitaka Shoji
Keyword(s):  
Regular Semigroup ◽  
Simple Semigroup ◽  
Finite Semigroup ◽  
Principal Factor ◽  
Regular Semigroups ◽  
Completely Simple Semigroup ◽  
Finite Semigroups ◽  
Completely Simple ◽  
Finite Regular Semigroup

In this paper, we prove that a completely 0-simple (or completely simple) semigroup is an amalgamation base for finite semigroups if and only if it is an amalgamation base for semigroups. By adopting the same method as used in a previous paper, we prove that a finite regular semigroup is an amalgamation base for finite semigroups if its [Formula: see text]-classes are linearly ordered and all of its principal factor semigroups are amalgamation bases for finite semigroups. Finally, we give an example of a finite semigroup U which is an amalgamation base for semigroups, but not all of its principal factor semigroups are amalgamation bases either for semigroups or for finite semigroups.

scholarly journals Congruence lattices of regular semigroups related to kernels and traces

1991 ◽  
Vol 34 (2) ◽  
pp. 179-203 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Distributive Lattice ◽  
Regular Semigroup ◽  
Homomorphic Image ◽  
Regular Semigroups ◽  
Generators And Relations ◽  
Congruence Lattices ◽  
Left And Right ◽  
Lattice Of Congruences

The kernel–trace approach to congruences on a regular semigroup S can be refined by introducing the left and right traces. This induces eight operators on the lattice of congruences on S: t1, k, tr,; Tt, K, Tr; t, T. We describe the lattice of congruences on S generated by six 3-element subsets of the set {ωt1, ωk, ωtr, εTt, εK, εTr} where ω and ε denote the universal and the equality relations. This is effected by means of a diagram and in terms of generators and relations on a free distributive lattice, or a homomorphic image thereof. We perform the same analysis for the lattice of congruences on S generated by the set {εK, ωk, εT, ωt}.

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