scholarly journals Congruences contained in an equivalence on a semigroup

1980 ◽  
Vol 29 (2) ◽  
pp. 162-176 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Subdirect Product ◽  
Congruence Lattices ◽  
Lattice Of Congruences

AbstractThe main theorem of this paper shows that the lattice of congruences contained is some equivalence π on a semigroup S can be decomposed into a subdirect product of sublattices of the congruence lattices on the ‘prinipal π-facotrsρ of S—the semigroups formed by adjoining zeroes to the π-classes—whenever these are well-defined. The theorem is then applied to various equavalences and classes of semigroups to give some new results and alternative proofs of known ones.

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 The congruence lattice of a combinatorial strict inverse semigroup

1994 ◽  
Vol 37 (1) ◽  
pp. 25-37 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Inverse Semigroup ◽  
Congruence Lattice ◽  
Subdirect Product ◽  
Congruence Lattices

The congruence lattice of a combinatorial strict inverse semigroup is shown to be isomorphic to a complete subdirect product of congruence lattices of semilattices preserving pseudocomplements.

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.

scholarly journals Congruence pairs of principal MS-algebras and perfect extensions

2020 ◽  
Vol 70 (6) ◽  
pp. 1275-1288
Author(s):  
Abd El-Mohsen Badawy ◽  
Miroslav Haviar ◽  
Miroslav Ploščica
Keyword(s):  
Boolean Algebra ◽  
Congruence Lattices ◽  
Descriptive Solution ◽  
The One ◽  
Special Case ◽  
Congruence Pair

AbstractThe notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

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.

scholarly journals Subdirect products of semirings

2001 ◽  
Vol 26 (9) ◽  
pp. 539-545
Author(s):  
P. Mukhopadhyay
Keyword(s):  
Distributive Lattice ◽  
Subdirect Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Subdirect Products ◽  
Necessary And Sufficient

Bandelt and Petrich (1982) proved that an inversive semiringSis a subdirect product of a distributive lattice and a ring if and only ifSsatisfies certain conditions. The aim of this paper is to obtain a generalized version of this result. The main purpose of this paper however, is to investigate, what new necessary and sufficient conditions need we impose on an inversive semiring, so that, in its aforesaid representation as a subdirect product, the “ring” involved can be gradually enriched to a “field.” Finally, we provide a construction of fullE-inversive semirings, which are subdirect products of a semilattice and a ring.

scholarly journals N-th root rings

1987 ◽  
Vol 35 (1) ◽  
pp. 111-123 ◽  
Author(s):  
Henry Heatherly ◽  
Altha Blanchet
Keyword(s):  
Commutative Ring ◽  
Subdirect Product ◽  
General Construction ◽  
Fixed Integer ◽  
Chain Conditions

A ring for which there is a fixed integer n ≥ 2 such that every element in the ring has an n-th in the ring is called an n-th root ring. This paper gives numerous examples of diverse types of n-th root rings, some via general construction procedures. It is shown that every commutative ring can be embedded in a commutative n-th root ring with unity. The structure of n-th root rings with chain conditions is developed and finite n-th root rings are completely classified. Subdirect product representations are given for several classes of n-th root rings.

scholarly journals CONGRUENCE FD-MAXIMAL VARIETIES OF ALGEBRAS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250053 ◽  
Author(s):  
PIERRE GILLIBERT ◽  
MIROSLAV PLOŠČICA
Keyword(s):  
Necessary Condition ◽  
Finitely Generated ◽  
Distributive Variety ◽  
Finite Lattices ◽  
Congruence Lattices ◽  
Congruence Distributive Variety ◽  
Congruence Distributive ◽  
Special Congruence

We study the class of finite lattices that are isomorphic to the congruence lattices of algebras from a given finitely generated congruence-distributive variety. If this class is as large as allowed by an obvious necessary condition, the variety is called congruence FD-maximal. The main results of this paper characterize some special congruence FD-maximal varieties.

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