scholarly journals A Note on Locally Inverse Semigroup Algebras

2008 ◽  
Vol 2008 ◽  
pp. 1-5 ◽  
Author(s):  
Xiaojiang Guo
Keyword(s):  
Inverse Semigroup ◽  
Commutative Ring ◽  
Direct Product ◽  
Necessary Conditions ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Locally Inverse Semigroup ◽  
Maximum Subgroup ◽  
Sufficient And Necessary Conditions

LetRbe a commutative ring andSa finite locally inverse semigroup. It is proved that the semigroup algebraR[S]is isomorphic to the direct product of Munn algebrasℳ(R[GJ],mJ,nJ;PJ)withJ∈S/𝒥, wheremJis the number ofℛ-classes inJ,nJthe number ofℒ-classes inJ, andGJa maximum subgroup ofJ. As applications, we obtain the sufficient and necessary conditions for the semigroup algebra of a finite locally inverse semigroup to be semisimple.

scholarly journals Self-injectivity of semigroup algebras

2020 ◽  
Vol 18 (1) ◽  
pp. 333-352
Author(s):  
Junying Guo ◽  
Xiaojiang Guo
Keyword(s):  
Regular Semigroup ◽  
Necessary Conditions ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Abundant Semigroup ◽  
Regular Semigroups ◽  
Abundant Semigroups ◽  
Sufficient And Necessary Conditions ◽  
Finite Regular Semigroup

Abstract It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K 0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S] is a right (respectively, left) self-injective imply that S is finite? (Semigroup Algebras, Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K 0[S] being right (left) self-injective when K 0[S] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.

Semiprimeness of semigroup algebras

2021 ◽  
Vol 19 (1) ◽  
pp. 803-832
Author(s):  
Junying Guo ◽  
Xiaojiang Guo
Keyword(s):  
Regular Semigroup ◽  
Regularity Condition ◽  
Necessary Conditions ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Abundant Semigroup ◽  
Locally Finite ◽  
Regular Semigroups ◽  
Abundant Semigroups ◽  
Sufficient And Necessary Conditions

Abstract Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and study D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices of a semigroup. Based on D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebra A {\mathcal{A}} with unity, A {\mathcal{A}} is primitive (prime) if and only if so is M n ( A ) {M}_{n}\left({\mathcal{A}}) . Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.

THE FIRST COHOMOLOGY GROUP OF BANACH INVERSE SEMIGROUP ALGEBRAS WITH COEFFICIENTS IN -EMBEDDED BANACH BIMODULES

2019 ◽  
Vol 101 (3) ◽  
pp. 488-495
Author(s):  
HOGER GHAHRAMANI
Keyword(s):  
Banach Space ◽  
Inverse Semigroup ◽  
Cohomology Group ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Mild Conditions ◽  
First Cohomology Group

Let $S$ be a discrete inverse semigroup, $l^{1}(S)$ the Banach semigroup algebra on $S$ and $\mathbb{X}$ a Banach $l^{1}(S)$-bimodule which is an $L$-embedded Banach space. We show that under some mild conditions ${\mathcal{H}}^{1}(l^{1}(S),\mathbb{X})=0$. We also provide an application of the main result.

scholarly journals Moore-Penrose inversion in complex contracted inverse semigroup algebras

1999 ◽  
Vol 66 (3) ◽  
pp. 297-302
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Group Algebra ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Locally Finite ◽  
Complex Group ◽  
Complex Group Algebra

AbstractIt is shown that every element of the complex contracted semigroup algebra of an inverse semigroup S = S0 has a Moore-Penrose inverse, with respect to the natural involution, if and only if S is locally finite. In particular, every element of a complex group algebra has such an inverse if and only if the group is locally finite.

Trace functions on the algebras of certain E-unitary inverse semigroups

1995 ◽  
Vol 125 (5) ◽  
pp. 1077-1084 ◽  
Author(s):  
M. J. Crabb ◽  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebra ◽  
Inverse Semigroups ◽  
Trace Function ◽  
Complex Field ◽  
Semigroup Algebras ◽  
Complex Conjugation ◽  
Arbitrary Rank

A construction is given for a trace function on the semigroup algebra of a certain type of E-unitary inverse semigroup over any subfield of the complex field that is closed under complex conjugation. In particular, the method applies to the semigroup algebras of free inverse semigroups of arbitrary rank.

scholarly journals Module amenability of restricted semigroup algebras under module actions

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 787-793
Author(s):  
Abbas Sahleh ◽  
Somaye Tanha
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Clifford Semigroup ◽  
Module Amenability

In this article, weshow that module amenability with the canonical action of restricted semigroup algebra l1r (S) and semigroup algebra l1(Sr) are equivalent, where Sr is the restricted semigroup of associated to the inverse semigroup S. We use this to give a characterization of module amenability of restricted semigroup algebra l1r (S) with the canonical action, where S is a Clifford semigroup.

scholarly journals Trace functions on inverse semigroup algebras

1995 ◽  
Vol 52 (3) ◽  
pp. 359-372 ◽  
Author(s):  
D. Easdown ◽  
W.D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Semigroup Algebra ◽  
Inverse Semigroups ◽  
Trace Function ◽  
Complex Field ◽  
Semigroup Algebras ◽  
Complex Conjugation ◽  
Necessary And Sufficient ◽  
Completely Semisimple

Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F[S], the semigroup algebra of S over F. Necessary and sufficient conditions on S are found for the existence of a trace function on F[S] that takes positive integral values on the idempotents of S. Although F[S] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S. This is used to show that the natural involution on F[S] is special. It also leads to the construction of a trace function on F[S] for the case in which F is the real or complex field and S is completely semisimple of a type that includes countable free inverse semigroups.

scholarly journals On Arc Connectivity of Direct-Product Digraphs

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Tiedan Zhu ◽  
Jianping Ou
Keyword(s):  
Explicit Expression ◽  
Direct Product ◽  
Necessary Conditions ◽  
Strongly Connected ◽  
Sufficient And Necessary Conditions

Four natural orientations of the direct product of two digraphs are introduced in this paper. Sufficient and necessary conditions for these orientations to be strongly connected are presented, as well as an explicit expression of the arc connectivity of a class of direct-product digraphs.

scholarly journals The equality of Hochschild cohomology group and module cohomology group for semigroup algebras

2022 ◽  
Vol 40 ◽  
pp. 1-9
Author(s):  
Ebrahim Nasrabadi
Keyword(s):  
Inverse Semigroup ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Hochschild Cohomology Group

‎Let $S$ be a commutative inverse semigroup with idempotent set $E$‎. ‎In this paper‎, ‎we show that for every $n\in \mathbb{N}_0$‎, ‎$n$-th Hochschild cohomology group of semigroup algebra $\ell^1(S)$ with coefficients in $\ell^\infty(S)$ and its $n$-th $\ell^1(E)$-module cohomology group‎, ‎are equal‎. ‎Indeed‎, ‎we prove that‎ ‎\[ \HH^{n}(\ell^1(S),\ell^\infty(S))=\HH^{n}_{\ell^1(E)}(\ell^1(S),\ell^\infty(S)),\] for all $n\geq 0$‎.

Sign in / Sign up

Export Citation Format

Share Document