Naturally ordered regular semigroups with a greatest idempotent

1981 ◽  
Vol 91 (1-2) ◽  
pp. 107-122 ◽  
Author(s):  
T. S. Blyth ◽  
R. McFadden
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Natural Order ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
Green's Relations ◽  
Structure Theorems ◽  
Necessary And Sufficient

SynopsisWe consider ordered regular semigroups in which the order extends the natural order on the idempotents, and which are graced with the presence of a greatest idempotent. This implies that every element has a greatest inverse. An investigation into the properties ofthese special elements allows a description of Green's relations on the subsemigroup generated by the idempotents. This in turn leads to a complete description of the structure of idempotent-generated naturally ordered regular semigroups having a greatest idempotent. The smallest such semigroup that is not orthodox is also described. These results lead us to obtain structure theorems in the general case with the added condition that Green's relations be regular. Finally, necessary and sufficient conditions for such a semigroup to be a Dubreil-Jacotin semigroup are found.

scholarly journals A NEW CONSTRUCTION FOR REGULAR SEMIGROUPS WITH QUASI-IDEAL ORTHODOX TRANSVERSALS

2009 ◽  
Vol 86 (2) ◽  
pp. 177-187 ◽  
Author(s):  
XIANGJUN KONG ◽  
XIANZHONG ZHAO
Keyword(s):  
Regular Semigroup ◽  
Structure Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
New Construction ◽  
Necessary And Sufficient ◽  
Equivalent Conditions ◽  
Orthodox Semigroups

AbstractIn any regular semigroup with an orthodox transversal, we define two sets R and L using Green’s relations and give necessary and sufficient conditions for them to be subsemigroups. By using R and L, some equivalent conditions for an orthodox transversal to be a quasi-ideal are obtained. Finally, we give a structure theorem for regular semigroups with quasi-ideal orthodox transversals by two orthodox semigroups R and L.

Fuzzy ∗–ideals and their applications in characterizing abundance and regularity of a semigroup

2021 ◽  
pp. 1-8
Author(s):  
Chunhua Li ◽  
Baogen Xu ◽  
Huawei Huang
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Arbitrary Semigroup ◽  
Green's Relations ◽  
Fuzzy Ideal ◽  
Necessary And Sufficient ◽  
The Relationship

In this paper, the notion of a fuzzy *–ideal of a semigroup is introduced by exploiting generalized Green’s relations L * and R * , and some characterizations of fuzzy *–ideals on an arbitrary semigroup are obtained. Our main purpose is to establish the relationship between fuzzy *–ideals and abundance for an arbitrary semigroup. As an application of our results, we also give some new necessary and sufficient conditions for an arbitrary semigroup to be regular and inverse, respectively.

scholarly journals Semigroups of left quotients—the uniqueness problem

1992 ◽  
Vol 35 (2) ◽  
pp. 213-226 ◽  
Author(s):  
Victoria Gould
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Uniqueness Problem ◽  
Green’S Relations ◽  
Multiplicative Semigroup ◽  
Cancellation Condition ◽  
Green's Relations ◽  
Necessary And Sufficient ◽  
Positive Results

Let S be a subsemigroup of a semigroup Q. Then Q is a semigroup of left quotients of S if every element of Q can be written as a*b, where a lies in a group -class of Q and a* is the inverse of a in this group; in addition, we insist that every element of S satisfying a weak cancellation condition named square-cancellable lie in a subgroup of Q.J. B. Fountain and M. Petrich gave an example of a semigroup having two non-isomorphic semigroups of left quotients. More positive results are available if we restrict the classes of semigroups from which the semigroups of left quotients may come. For example, a semigroup has at most one bisimple inverse ω-semigroup of left quotients. The crux of the matter is the restrictions to a semigroup S of Green's relations ℛ and ℒ in a semigroup of quotients of S. With this in mind we give necessary and sufficient conditions for two semigroups of left quotients of S to be isomorphic under an isomorphism fixing S pointwise.The above result is then used to show that if R is a subring of rings Q1 and Q2 and the multiplicative subsemigroups of Q1 and Q2 are semigroups of left quotients of the multiplicative semigroup of R, then Ql and Q2 are isomorphic rings.

scholarly journals Eventually regular semigroups that are group-bound

1986 ◽  
Vol 34 (1) ◽  
pp. 127-132 ◽  
Author(s):  
P. M. Edwards
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Regular Semigroups ◽  
Necessary And Sufficient

Necessary and sufficient conditions are given for certain classes of eventually regular semigroups to the group-bound or even periodic.

scholarly journals Eventually Pointed Principally Ordered Regular Semigroups

2020 ◽  
Vol 24 (2) ◽  
pp. 139
Author(s):  
G.A. Pinto
Keyword(s):  
Positive Integer ◽  
Regular Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Completely Simple

An ordered regular semigroup, , is said to be principally ordered if for every  there exists . A principally ordered regular semigroup is pointed if for every element,  we have . Here we investigate those principally ordered regular semigroups that are eventually pointed in the sense that for all  there exists a positive integer, , such that . Necessary and sufficient conditions for an eventually pointed principally ordered regular semigroup to be naturally ordered and to be completely simple are obtained. We describe the subalgebra of  generated by a pair of comparable idempotents  and such that . 

Twisted Brauer monoids

2018 ◽  
Vol 148 (4) ◽  
pp. 731-750 ◽  
Author(s):  
Igor Dolinka ◽  
James East
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Singular Part ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Generating Set ◽  
Idempotent Rank ◽  
Singular Ideal ◽  
Necessary And Sufficient ◽  
Lattice Of Ideals

We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.

scholarly journals The Direct Decomposition of l-algebras into Products of Subdirectly Irreducible Factors

2003 ◽  
Vol 75 (1) ◽  
pp. 41-56 ◽  
Author(s):  
Sándor Radeleczki
Keyword(s):  
Direct Product ◽  
Present Author ◽  
Sufficient Conditions ◽  
Direct Decomposition ◽  
Necessary And Sufficient Conditions ◽  
Subdirectly Irreducible ◽  
Bounded Lattice ◽  
Structure Theorems ◽  
Necessary And Sufficient

AbstractGeneralizing earlier results of Katriňák, El-Assar and the present author we prove new structure theorems for l-algebras. We obtain necessary and sufficient conditions for the decomposition of an arbitrary bounded lattice into a direct product of (finitely) subdirectly irreducible lattices.

On the Structure of Inverse Semigroup Amalgams

1997 ◽  
Vol 07 (05) ◽  
pp. 577-604 ◽  
Author(s):  
Paul Bennett
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Maximal Subgroups ◽  
Regular Semigroups ◽  
Endomorphism Monoids ◽  
Amalgamated Free Product ◽  
Necessary And Sufficient

This paper is the second of two papers devoted to the study of amalgamated free products of inverse semigroups. We use the characterization of the Schützenberger automata given previously by the author to obtain structural results and preservational properties of lower bounded amalgams. Haataja, Margolis and Meakin have shown that if [S1,S2;U is an amalgam of regular semigroups in which S1∩ S2=U is a full regular subsemigroup of S1 and S2, then the maximal subgroups of the amalgamated free product S1*U S2 may be described by the fundamental groups of certain bipartite graphs of groups. In this paper we show that the maximal subgroups of a lower bounded amalgam [S1,S2;U] are either isomorphic copies of subgroups of S1 and S2 or can be described by the same Bass-Serre theory characterization. It follows, as for the regular case, that if S1 and S2 are combinatorial, then the maximal subgroups of S1*U S2 are free. By studying the endomorphism monoids of the Schützenberger graphs we obtain a number of results concerning when inverse semigroup properties are preserved under the amalgamated free product construction. For example, necessary and sufficient conditions are given for S1*U S2 to be completely semisimple. Under a mild assumption we establish necessary and sufficient conditions for S1*U S2 to have finite ℛ-classes. This enables us to reprove a result of Cherubini, Meakin and Piochi on amalgams of free inverse semigroups. Finally we give sufficient conditions for S1*U S2 to be E-unitary.

scholarly journals The kernel relation for a strict extension of certain regular semigroups

1996 ◽  
Vol 38 (3) ◽  
pp. 347-357 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Congruence Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Regular Semigroups ◽  
Strict Extension ◽  
Necessary And Sufficient ◽  
Special Classes

Let R be a regular semigroup and denote by (R) its congruence lattice. For , the kernel of pis the set ker . The relation K on (R) defined by λKp if ker λ = ker p is the kernel relation on (R). In general, K is a complete ∩-congruence but it is not a v-congruence. In view of the importance of the kernel-trace approach to the study of congruences on a regular semigroup (the trace of p is its restriction to idempotents of R), it is of considerable interest to determine necessary and sufficient conditions on R in order for K to be a congruence. This being in general a difficult task, one restricts attention to special classes of regular semigroups. For a background on this subject, consult [1].

scholarly journals On Left Bipotent Near-Rings

1979 ◽  
Vol 22 (2) ◽  
pp. 99-107 ◽  
Author(s):  
J. L. Jat ◽  
S. C. Choudhary
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Singular Set ◽  
Direct Sum ◽  
Zero Divisors ◽  
Structure Theorems ◽  
Necessary And Sufficient ◽  
Regular Section

A near-ring N is defined to be left bipotent if Na = Na2 for each a in N. Many properties of such near-rings are proved in Section 1, and results of Chandran (4) are generalised. Most of the results are different from, and contrary to, the ring case. Necessary and sufficient conditions have also been obtained under which such near-rings become regular. Section 2 deals with left bipotent near-rings without zero divisors. Some structure theorems for direct sum decompositions and J(N) = (0) are proved and it is shown that for a left bipotent S-near-ring, the singular ‘set’ S(N) = 0. Necessary examples and counter examples are supplied.

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