scholarly journals Minimum Group Congruences on Eventually Regular Semigroups

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-5
Author(s):  
Wang Yu ◽  
Yin ZhiXiang
Keyword(s):  
Regular Semigroup ◽  
Group Congruence ◽  
Regular Semigroups ◽  
Minimum Group ◽  
Eventually Regular Semigroup

An eventually regular semigroup is a semigroup in which some power of any element is regular. The minimum group congruence on an eventually regular semigroup is investigated by means of weak inverse. Furthermore, some properties of the minimum group congruence on an eventually regular semigroup are characterized.

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.

scholarly journals Congruences and Green's relations on eventually regular semigroups

1987 ◽  
Vol 43 (1) ◽  
pp. 64-69 ◽  
Author(s):  
P. M. Edwards ◽  
T. E. Hall
Keyword(s):  
Regular Semigroup ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
Regular Elements ◽  
Green's Relations ◽  
Eventually Regular Semigroup

AbstractA semigroup is eventually regular if each of its elements has some power that is regular. Let 𝓚 be one of Green's relations and let ρ be a congruence on an eventually regular semigroup S. It is shown for 𝓚 = 𝓛, 𝓡 and 𝓓 that if A and B are regular elements of S/ρ that are 𝓚-related in S/ρ then there exist elements a ∈ A, b ∈ B such that a and b are 𝓚-related in S. The result is not true for 𝓗 or 𝓙.

scholarly journals Certain congruences on a completely regular semigroup

1974 ◽  
Vol 15 (2) ◽  
pp. 109-120 ◽  
Author(s):  
Thomas L. Pirnot
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Group Congruence ◽  
Completely Regular Semigroup ◽  
Rees Matrix Semigroup ◽  
Semilattice Congruence ◽  
Completely Regular ◽  
Maximal Group ◽  
Minimum Group ◽  
Matrix Semigroup

Congruences on a semigroup S such that the corresponding factor semigroups are of a special type have been considered by several authors. Frequently it has been difficult to obtain worthwhile results unless restrictions have been imposed on the type of semigroup considered. For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences ρ on a completely regular semigroup S such that S/ρ is a semilattice of groups. We shall call such a congruence an SG-congruence.

scholarly journals Certain fundamental congruences on a regular semigroup

1966 ◽  
Vol 7 (3) ◽  
pp. 145-159 ◽  
Author(s):  
J. M. Howie ◽  
G. Lallement
Keyword(s):  
Inverse Semigroup ◽  
Algebraic Theory ◽  
Primary Concern ◽  
Group Congruence ◽  
Semilattice Congruence ◽  
Regular Semigroups ◽  
Recent Developments ◽  
Minimum Group ◽  
Band Congruence ◽  
Lattice Of Congruences

In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some restriction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating congruence on a regular or inverse semigroup by the authors separately [9, 10] and by Munn [14], and the minimum semilattice congruence on a general or commutative semigroup by Tamura and Kimura [19], Yamada [22], Clifford [3] and Petrich [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall consistently denote by α β and η respectively.

Certain congruences on eventually regular semigroups I

2015 ◽  
Vol 52 (4) ◽  
pp. 434-449
Author(s):  
Roman S. Gigoń
Keyword(s):  
Regular Semigroup ◽  
Orthodox Semigroup ◽  
Full Description ◽  
Clifford Semigroup ◽  
Regular Semigroups ◽  
Subdirect Products ◽  
Eventually Regular Semigroup

A semigroup is called eventually regular if each of its elements has a regular power. In this paper we study certain fundamental congruences on an eventually regular semigroup. We generalize some results of Howie and Lallement (1966) and LaTorre (1983). In particular, we give a full description of the semilattice of group congruences (together with the least such a congruence) on an arbitrary eventually regular (orthodox) semigroup. Moreover, we investigate UBG-congruences on an eventually regular semigroup. Finally, we study the eventually regular subdirect products of an E-unitary semigroup and a Clifford semigroup.

R-Unipotent Congruences on Eventually Regular Semigroups

2007 ◽  
Vol 14 (01) ◽  
pp. 37-52 ◽  
Author(s):  
Yanfeng Luo ◽  
Xiaoling Li
Keyword(s):  
Positive Integer ◽  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Regular Semigroups ◽  
Normal Subsemigroup ◽  
Normal Congruence ◽  
Eventually Regular Semigroup

A semigroup S is called an eventually regular semigroup if for every a ∈ S, there exists a positive integer n such that an is regular. In this paper, the R-unipotent, inverse semigroup and group congruences on an eventually regular semigroup S are described by means of certain congruence pairs (ξ, K), where ξ is a normal congruence on the subsemigroup 〈E(S)〉 generated by E(S), and K is a normal subsemigroup of S.

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.

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