Relations on the lattice of congruences of a regular semigroup

1985 ◽  
Vol 31 (1) ◽  
pp. 227-233
Author(s):  
R. J. Koch ◽  
B. L. Madison
Keyword(s):  
Regular Semigroup ◽  
Lattice Of Congruences

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 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 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 Congruences on an orthodox semigroup via the minimum inverse semigroup congruence

1977 ◽  
Vol 18 (2) ◽  
pp. 181-192 ◽  
Author(s):  
Carl Eberhart ◽  
Wiley Williams
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Orthodox Semigroup ◽  
Band Congruence ◽  
Lattice Of Congruences

It is well known that the lattice Λ(S) of congruences on a regular semigroup S contains certain fundamental congruences. For example there is always a minimum band congruence β, which Spitznagel has used in his study of the lattice of congruences on a band of groups [16]. Of key importance to his investigation is the fact that β separates congruences on a band of groups in the sense that two congruences are the same if they have the same meet and join with β. This result enabled him to characterize θ-modular bands of groups as precisely those bands of groups for which ρ⃗(ρ∨β, ρ∧β)is an embedding of Λ(S) into a product of sublattices.

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.

A certain sublattice of the lattice of congruences on a regular semigroup

1964 ◽  
Vol 60 (3) ◽  
pp. 385-391 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Regular Semigroup ◽  
Left Ideal ◽  
Modular Lattice ◽  
Partial Ordering ◽  
Lattice Of Congruences

It is well known that, with respect to the natural partial ordering, the set of all congruences on a group forms a modular lattice. In the present paper we develop an extension of this result to the case of a regular semigroup S (α ∈ αSα for all α in S). Let Σ(ℋ) denote the set of all congruences on S with the property that congruent elements generate the same principal left ideal and the same principal right ideal of S. We show (Theorem 1) that, under the natural partial ordering, Σ(ℋ) is a modular lattice with a greatest and a least element.

On Lattice-Ordered Rees Matrix Γ-Semigroups

2013 ◽  
Vol 59 (1) ◽  
pp. 209-218 ◽  
Author(s):  
Kostaq Hila ◽  
Edmond Pisha
Keyword(s):  
Structure Theorem ◽  
Lattice Of Congruences

Abstract The purpose of this paper is to introduce and give some properties of l-Rees matrix Γ-semigroups. Generalizing the results given by Guowei and Ping, concerning the congruences and lattice of congruences on regular Rees matrix Γ-semigroups, the structure theorem of l-congruences lattice of l - Γ-semigroup M = μº(G : I; L; Γe) is given, from which it follows that this l-congruences lattice is distributive.

XII.—The Lattice of Congruences on a Bisimple ω-Semigroup

1967 ◽  
Vol 67 (3) ◽  
pp. 175-184
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Natural Numbers ◽  
Necessary And Sufficient ◽  
Lattice Of Congruences

SynopsisA necessary and sufficient condition is determined for the modularity of the lattice of congruences on a bisimple inverse semigroup whose semilattice of idempotents is order-anti-isomorphic to the set of natural numbers.

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