A construction of commutative nilsemigroups
type: Item
instance of: scholarly article
title: A construction of commutative nilsemigroups
zbMATH Open document ID: 1006.20046
DOI: 10.1023/A:1014967701458
author: Pierre Antoine Grillet
published in: Periodica Mathematica Hungarica
publication date: 2003-02-18T00:00:00Z
review text: A commutative nilsemigroup \(S\) is a commutative semigroup with zero 0 and for every element \(a\in S\) there is \(n\in\mathbb{Z}_+\) such that \(a^n=0\). In this paper the author constructs all nilsemigroups by means of the quotient semigroup of a free commutative semigroup \(F\) using the concept of ``coherent nests'' of equivalence relations on some set. This paper extends the author's earlier results [Commun. Algebra 19, No. 11, 3145-3172 (1991; Zbl 0755.20016)]. Let \(S\) be a semigroup, \(I\) an ideal of \(S\), and \(C\) a congruence on \(S\). If \(I\) is a \(C\)-class, then \(C\) is called \(I\)-congruence on \(S\). A nilsemigroup ideal \(I\) of \(S\) is an ideal of \(S\) such that \(S/I\) is a nilsemigroup. A coideal \(H\) of \(S\) is a subset \(H\) of \(S\) such that \(S\setminus H\) is an ideal of \(S\). In particular if \(H\) is a coideal of \(S\) and \(S\setminus H\) is a nilsemigroup ideal, then \(H\) is called a nilsemigroup coideal of \(S\). Let \(K\) denote the set of all maximal elements of \(H\). We assume that \(H\) is generated by its maximal elements. Given an \(I\)-congruence \(C\) and for every \(u\in H\) we define an equivalence relation \(C_u\) on \(K\) as follows: \[ kC_u\ell\text{ if and only if }k=\ell\text{ or }k,\ell\geq u\text{ and }k-uC\ell-u. \] The family \(N(C)=(C_u)_{u\in H}\) is called the nest of equivalence relations of \(C\). When \(H\) is generated by its maximal elements, every \(C_u\) is an equivalence relation on \(K\) and the following conditions are satisfied: for \(u,v\in H\), (N1) \(u\leq v\Rightarrow C_v\subseteq C_u\), (N2) \(C_u\subseteq D_u\) where \(aD_ub\) iff \(K:a=K:b\) in the sense of \textit{R. Croisot} [J. Math. Pure Appl., IX. Sér. 36, 373-417 (1957; Zbl 0079.25002)], (N3) if \(k,\ell,m,n\in K\) and \(k-\ell=m-n\), then for every \(u\in H\), \(kC_{p+u}\ell\iff mC_{q+u}n\) where \(p=k-k\land m\), \(q=m-k\land m\). A family \((C_u)_{u\in H}\) of equivalence relations on \(K\) satisfying (N1), (N2) and (N3) i…
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 1870464
zbMATH Keywords: commutative nilsemigroups
zbMATH Keywords: free semigroups
zbMATH Keywords: coherent nests of equivalence relations
zbMATH Keywords: nilsemigroup ideals
zbMATH Keywords: coideals
zbMATH Keywords: congruences
reviewed by: Takayuki Tamura
MaRDI profile type: Publication
full work available at URL: a:1014967701458
OpenAlex ID: W1812845037
description: scientific article; zbMATH DE number 1870464