The Chain of Ideals of Regular Semigroup Algebras

志荣 苏

doi:10.12677/pm.2021.1112231

Self-injectivity of semigroup algebras

Open Mathematics ◽

10.1515/math-2020-0023 ◽

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.

Download Full-text

On regular semigroup rings

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500026020 ◽

1984 ◽

Vol 99 (1-2) ◽

pp. 145-151 ◽

Cited By ~ 10

Author(s):

Jan Okniński

Keyword(s):

Regular Semigroup ◽

Semigroup Algebra ◽

Semigroup Algebras ◽

Semigroup Rings ◽

Principal Ideals ◽

Algebra Over A Field

SynopsisIt is shown that if A is an algebra over a field, then the regularity of the semigroup algebra A[G] implies that the semigroup G is periodic. This enables us to characterize regular semigroup algebras of semigroups with d.c.c. on principal ideals. Also, regular self-injective semigroup algebras are described.

Download Full-text

Semiprimeness of semigroup algebras

Open Mathematics ◽

10.1515/math-2021-0026 ◽

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.

Download Full-text

Spectra in Some Inverse Semigroup Algebras

Mathematical Proceedings of the Royal Irish Academy ◽

10.3318/pria.2004.104.2.211 ◽

2004 ◽

Vol 104 (2) ◽

pp. 211-218 ◽

Cited By ~ 1

Author(s):

M. J. Crabb ◽

J. Duncan ◽

C. M. McGregor

Keyword(s):

Inverse Semigroup ◽

Semigroup Algebras

Download Full-text

Short Generating Functions for some Semigroup Algebras

The Electronic Journal of Combinatorics ◽

10.37236/1729 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 23

Author(s):

Graham Denham

Keyword(s):

Rational Function ◽

Generating Functions ◽

Hilbert Series ◽

Simple Expression ◽

Arbitrary Field ◽

Graded Algebra ◽

Semigroup Algebras ◽

Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

Download Full-text