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 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 On the lattice of congruences on an eventually regular semigroup

1985 ◽  
Vol 38 (2) ◽  
pp. 281-286 ◽  
Author(s):  
P. M. Edwards
Keyword(s):  
Regular Semigroup ◽  
Subject Classification ◽  
Maximum Element ◽  
M 10 ◽  
Natural Equivalence ◽  
20 M 10 ◽  
Complete Sublattice ◽  
Eventually Regular Semigroup ◽  
Lattice Of Congruences

AbstractA natural equivalence θ on the lattice of congruences λ(S) of a semigroup S is studied. For any eventually regular semigroup S, it is shown that θ is a congruence, each θ-class is a complete sublattice of λ(S) and the maximum element in each θ-class is determined. 1980 Mathematics subject classification (Amer. Math. Soc.): 20 M 10.

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}.

scholarly journals On the lattice of congruences on a regular semigroup

1969 ◽  
Vol 1 (2) ◽  
pp. 231-235 ◽  
Author(s):  
T. E. Hall
Keyword(s):  
Regular Semigroup ◽  
Complete Lattice ◽  
Inverse Semigroups ◽  
Natural Homomorphism ◽  
Lattice Homomorphism ◽  
Regular Semigroups ◽  
Lattice Of Congruences

A result of Reilly and Scheiblich for inverse semigroups is proved true also for regular semigroups. For any regular semigroup S the relation θ is defined on the lattice, Λ(S), of congruences on S by: (ρ, τ) ∈ θ if ρ and τ induce the same partition of the idempotents of S. Then θ is a congruence on Λ(S), Λ(S)/θ is complete and the natural homomorphism of Λ(S) onto Λ(S)/θ is a complete lattice homomorphism.

scholarly journals Free completely regular semigroups

1984 ◽  
Vol 25 (2) ◽  
pp. 241-254 ◽  
Author(s):  
P. G. Trotter
Keyword(s):  
Regular Semigroup ◽  
Structure Theory ◽  
Completely Regular Semigroup ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Additional Operation ◽  
Alternative Characterization ◽  
Completely Regular Semigroups

A completely regular semigroup is a semigroup that is a union of groups. The aim here is to provide an alternative characterization of the free completely regular semigroup Fcrx on a set X to that given by J. A. Gerhard in [3, 4].Although the structure theory for completely regular semigroups was initiated in 1941 [1] by A. H. Clifford it was not until 1968 that it was shown by D. B. McAlister [5] that Fcrx exists. More recently, in [7], M. Petrich demonstrated the existence of Fcrx by showing that completely regular semigroups form a variety of unary semigroups (that is, semigroups with the additional operation of inversion).

scholarly journals Normal partitions of idempotents of regular semigroups

1978 ◽  
Vol 26 (1) ◽  
pp. 110-114 ◽  
Author(s):  
P. G. Trotter
Keyword(s):  
Regular Semigroup ◽  
Subject Classification ◽  
Regular Semigroups ◽  
M 10 ◽  
20 M 10

AbstractA characterization is provided here for any normal partition of the set of idempotents of a regular semigroup S. As a by-product of the method used, a new characterization of the greatest congruence on S corresponding to a given normal partition of its idempotents is obtained.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 M 10.

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 congruence and quotient lattices related to completely 0-simple and primitive regular semigroups

1969 ◽  
Vol 10 (1) ◽  
pp. 21-24 ◽  
Author(s):  
H. E. Scheiblich
Keyword(s):  
Regular Semigroup ◽  
Simple Proof ◽  
Simple Semigroup ◽  
Chain Condition ◽  
Regular Semigroups ◽  
Lattice Of Congruences

G. Lallement [4] has shown that the lattice of congruences, Λ(S), on a completely 0-simple semigroupSis semimodular, thus improving G. B. Preston's result [5] that such a lattice satisfies the Jordan-Dedekind chain condition. More recently, J. M. Howie [2] has given a new and more simple proof of Lallement's result using work due to Tamura [9]. The purpose of this note is to extend the semimodularity result to primitive regular semigroups, to establish a theorem relating certain congruence and quotient lattices, and to provide a theorem for congruences on any regular semigroup.

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.

V-regular semigroups

1981 ◽  
Vol 88 (3-4) ◽  
pp. 275-291 ◽  
Author(s):  
K. S. S. Nambooripad ◽  
F. Pastijn
Keyword(s):  
Regular Semigroup ◽  
Transformation Semigroup ◽  
Inverse Semigroups ◽  
Full Transformation ◽  
Regular Semigroups ◽  
Von Neumann ◽  
Biordered Set ◽  
Von Neumann Regular ◽  
Strongly Regular

SynopsisA regular semigroup S is called V-regular if for any elements a, b ∈ S and any inverse (ab)′ of ab, there exists an inverse a′ of a and an inverse b′ of b such that (ab)′ = b′a′. A characterization of a V-regular semigroup is given in terms of its partial band of idempotents. The strongly V-regular semigroups form a subclass of the class of V-regular semigroups which may be characterized in terms of their biordered set of idempotents. It is shown that the class of strongly V-regular semigroups comprises the elementary rectangular bands of inverse semigroups (including the completely simple semigroups), a special class of orthodox semigroups (including the inverse semigroups), the strongly regular Baer semigroups (including the semigroups that are the multiplicative semigroup of a von Neumann regular ring), the full transformation semigroup on a set, and the semigroup of all partial transformations on a set.

Sign in / Sign up

Export Citation Format

Share Document