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 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.

Congruences on glrac semigroups (I)

2021 ◽  
pp. 2250240
Author(s):  
Haijun Liu ◽  
Xiaojiang Guo
Keyword(s):  
Complete Lattice ◽  
Lattice Homomorphism ◽  
Complete Sublattice ◽  
Restriction Semigroup ◽  
Lattice Of Congruences

The theory of congruences on semigroups is an important part in the theory of semigroups. The aim of this paper is to study [Formula: see text]-congruences on a glrac semigroup. It is proved that the [Formula: see text]-congruences on a glrac semigroup become a complete sublattice of its lattice of congruences. Especially, the structures of left restriction semigroup [Formula: see text]-congruences and the projection-separating [Formula: see text]-congruences on a glrac semigroup are established. Also, we demonstrate that they are both complete sublattice of [Formula: see text]-congruences and consider their relations with respect to complete lattice homomorphism.

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 Epis are onto for finite regular semigroups

1983 ◽  
Vol 26 (2) ◽  
pp. 151-162 ◽  
Author(s):  
T. E. Hall ◽  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Open Problem ◽  
Inverse Semigroups ◽  
Regular Semigroups ◽  
Preliminary Results ◽  
Absolutely Closed ◽  
Finite Regular Semigroup

After preliminary results and definitions in Section 1, we show in Section 2 that any finite regular semigroup is saturated, in the sense of Howie and Isbell [8] (that is, the dominion of a finite regular semigroup U in a strictly containing semigroup S is never S). This is equivalent of course to showing that in the category of semigroups any epi from a finite regular semigroup is in fact onto. Note for inverse semigroups the stronger result, that any inverse semigroup is absolutely closed [11, Theorem VII. 2.14] or [8, Theorem 2.3]. Further, any inverse semigroup is in fact an amalgamation base in the class of semigroups [10], in the sense of [5]. These stronger results are known to be false for finite regular semigroups [8, Theorem 2.9] and [5, Theorem 25]. Whether or not every regular semigroup is saturated is an open problem.

scholarly journals The structure mappings on a regular semigroup

1978 ◽  
Vol 21 (2) ◽  
pp. 135-142 ◽  
Author(s):  
John Meakin
Keyword(s):  
Regular Semigroup ◽  
Inverse Semigroups ◽  
Regular Band ◽  
Regular Semigroups ◽  
Biordered Set

In (5) the author showed how to construct all inverse semigroups from their trace and semilattice of idempotents: the construction is by means of a family of mappings between ℛ-classes of the semigroup which we refer to as the structure mappings of the semigroup. In (7) (see also (8) and (9)) K. S. S. Nambooripad has adopted a similar approach to the structure of regular semigroups: he shows how to construct regular semigroups from their trace and biordered set of idempotents by means of a family of mappings between ℛ-classes and between ℒ-classes of the semigroup which we again refer to as the structure mappings of the semigroup. In the present paper we aim to provide a simpler set of axioms characterising the structure mappings on a regular semigroup than the axioms (R1)-(R7) of Nambooripad (9). Two major differences occur between Nambooripad's approach (9) and the approach adopted here: first, we consider the set of idempotents of our semigroups to be equipped with a partial regular band structure (in the sense of Clifford (3)) rather than a biorder structure, and second, we shall enlarge the set of structure mappings used by Nambooripad.

scholarly journals The natural partial order on a regular semigroup

1980 ◽  
Vol 23 (3) ◽  
pp. 249-260 ◽  
Author(s):  
K. S. Subramonian Nambooripad
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Partial Order ◽  
Inverse Semigroups ◽  
Natural Partial Order ◽  
Simple Description ◽  
Regular Semigroups ◽  
Simple Congruence ◽  
Pseudo Inverse ◽  
Completely Simple

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

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.

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.

scholarly journals Coextensions of pseudo-Inverse semigroups by rectangular bands

1980 ◽  
Vol 30 (1) ◽  
pp. 73-86 ◽  
Author(s):  
John Meakin
Keyword(s):  
Regular Semigroup ◽  
Rectangular Band ◽  
Inverse Semigroups ◽  
Regular Semigroups ◽  
Pseudo Inverse

AbstractWe say that a regulär semigroup S is a coetension of a (regular) semigroup T by rectangular bands if there is a homomorphism ϕ: S → T from S onto T such that, for each e = e2 ∈ S, e(ϕ ϕ-1) is a rectangular band. Regular semigroups which are coextesions of pseudo-inverse semigroups by rectangular bands may be characterized as those regular semigroups S with the property that, for each e = e2 ∈ S, ω(e) = {f = f2 ∈ S: ef = f} and ωl(e) = {f = f2 ∈ S: fe = f} are bands: this paper is concerned with a study of such semigroups.

COMPLETE LATTICE HOMOMORPHISM OF STRONGLY REGULAR CONGRUENCES ON -INVERSIVE SEMIGROUPS

2015 ◽  
Vol 100 (2) ◽  
pp. 199-215 ◽  
Author(s):  
XINGKUI FAN ◽  
QIANHUA CHEN ◽  
XIANGJUN KONG
Keyword(s):  
Complete Lattice ◽  
Open Problem ◽  
Lattice Homomorphism ◽  
Regular Semigroups ◽  
Abstract Characterization ◽  
Strongly Regular ◽  
Left And Right ◽  
Regular Congruence

In this paper, we investigate strongly regular congruences on $E$-inversive semigroups $S$. We describe the complete lattice homomorphism of strongly regular congruences, which is a generalization of an open problem of Pastijn and Petrich for regular semigroups. An abstract characterization of left and right traces for strongly regular congruences is given. The strongly regular (sr) congruences on $E$-inversive semigroups $S$ are described by means of certain strongly regular congruence triples $({\it\gamma},K,{\it\delta})$ consisting of certain sr-normal equivalences ${\it\gamma}$ and ${\it\delta}$ on $E(S)$ and a certain sr-normal subset $K$ of $S$. Further, we prove that each strongly regular congruence on $E$-inversive semigroups $S$ is uniquely determined by its associated strongly regular congruence triple.

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