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.

Arithmetical Semigroup Rings

1980 ◽  
Vol 32 (6) ◽  
pp. 1361-1371 ◽  
Author(s):  
Bonnie R. Hardy ◽  
Thomas S. Shores
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Forthcoming Paper ◽  
Semigroup Ring ◽  
Necessary And Sufficient Conditions ◽  
Semigroup Rings ◽  
Principal Ideal Ring ◽  
Arithmetical Semigroup ◽  
Necessary And Sufficient

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4].

Regular, Commutative, Maximal Semigroups of Quotients

1975 ◽  
Vol 18 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Jurgen Rompke
Keyword(s):  
Commutative Semigroup ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Natural Question ◽  
Commutative Case ◽  
Ring Of Quotients ◽  
Singular Ideal ◽  
Necessary And Sufficient

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

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.

Study of Semi-projective Retractable Modules

2007 ◽  
Vol 14 (03) ◽  
pp. 489-496 ◽  
Author(s):  
A. Haghany ◽  
M. R. Vedadi
Keyword(s):  
Endomorphism Ring ◽  
Semiprime Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Singular Ideal ◽  
The Right ◽  
Necessary And Sufficient

For a semi-projective retractable module MR with endomorphism ring S, we prove u.dim MR= u.dim SS, and find necessary and sufficient conditions on M in order that S be respectively semiprime, right nonsingular, finitely cogenerated, cocyclic, or weakly co-Hopfian. Precise descriptions of the right singular ideal of S and the socle of M are given, and in addition if S is a semiprime ring, it is shown that MR is FI-extending if and only if SS is FI-extending.

scholarly journals On solvability of the matrix equation AXB = C over a principal ideal domain

2020 ◽  
pp. 47-50
Author(s):  
Volodymyr Prokip
Keyword(s):  
Matrix Equation ◽  
Principal Ideal ◽  
Normal Forms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
The Matrix ◽  
Smith Normal Forms ◽  
Necessary And Sufficient

In this paper we present conditions of solvability of the matrix equation AXB = B over a principal ideal domain. The necessary and sufficient conditions of solvability of equation AXB = B in term of the Smith normal forms and in term of the Hermi-te normal forms of matrices constructed in a certain way by using the coefficients of this equation are proposed. If a solution of this equation exists we propose the method for its construction.

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 FINITENESS CONDITIONS OF S-COHN–JORDAN EXTENSIONS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250063
Author(s):  
JERZY MATCZUK
Keyword(s):  
Representation Theory ◽  
Principal Ideal ◽  
Sufficient Conditions ◽  
Ring Theory ◽  
Necessary And Sufficient Conditions ◽  
Finiteness Conditions ◽  
Principal Ideal Ring ◽  
Ideal Ring ◽  
Necessary And Sufficient

Let a monoid S act on a ring R by injective endomorphisms and A(R; S) denote the S-Cohn–Jordan extension of R. Some results relating finiteness conditions of R and that of A(R; S) are presented. In particular necessary and sufficient conditions for A(R; S) to be left noetherian, to be left Bézout and to be left principal ideal ring are presented. This also offers a solution to Problem 10 from [On S-Cohn–Jordan extensions, in Proc. 39th Symp. Ring Theory and Representation Theory, Hiroshima, ed. M. Kutami (Hiroshima Univ., Japan, 2007), pp. 30–35].

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 B-almost distributive fuzzy lattice

2018 ◽  
Vol 47 (3) ◽  
Author(s):  
Berhanu Assaye ◽  
Mihret Alemneh ◽  
Gerima Tefera
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fuzzy Algebra ◽  
Fuzzy Lattice ◽  
Necessary And Sufficient

The paper introduces the concept of B-Almost distributive fuzzy lattice (BADFL) in terms of its principal ideal fuzzy lattice. Necessary and sufficient conditions for an ADFL to become a B-ADFL are investigated. We also prove the equivalency of B-algebra and B-fuzzy algebra. In addition, we extend PSADL to PSADFL and prove that B-ADFL implies PSADFL.

scholarly journals Dedekind’s Criterion and Integral Bases

2019 ◽  
Vol 73 (1) ◽  
pp. 1-8
Author(s):  
Lhoussain El Fadil
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Irreducible Polynomial ◽  
Integral Closure ◽  
Necessary And Sufficient Conditions ◽  
Principal Ideal Domain ◽  
Quotient Field ◽  
Ideal Domain ◽  
Integral Bases ◽  
Necessary And Sufficient

Abstract Let R be a principal ideal domain with quotient field K, and L = K(α), where α is a root of a monic irreducible polynomial F (x) ∈ R[x]. Let ℤL be the integral closure of R in L. In this paper, for every prime p of R, we give a new efficient version of Dedekind’s criterion in R, i.e., necessary and sufficient conditions on F (x) to have p not dividing the index [ℤL: R[α]], for every prime p of R. Some computational examples are given for R = ℤ.

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