Commutative zeropotent semigroups with few prime ideals.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1249.20061
author: Tomáš Kepka
author: Jaroslav Ježek
author: Petr Němec
publication date: 2012-07-10T00:00:00Z
full work available at URL: https://eudml.org/doc/246701
full work available at URL: cmuc1104.htm
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 6054238
zbMATH Keywords: infinite commutative semigroups
zbMATH Keywords: zeropotent semigroups
zbMATH Keywords: prime ideals
zbMATH Keywords: endomorphism monoids
MaRDI profile type: Publication
title: Commutative zeropotent semigroups with few prime ideals.
published in: Commentationes Mathematicae Universitatis Carolinae
review text: This paper is a contribution to the open problem: Is there an infinite commutative semigroup with a finite endomorphism monoid? Let \(S\) be a commutative semigroup satisfying the identity \(x+x=y+y+y\), a subset \(I\subseteq S\) is an ideal containing \(0\) (\(0\) is a zero of \(S\), hence \(0=x+x\) for some \(x\in S\)) and \(x+y\in I\) whenever \(x\in I\) or \(y\in I\). An ideal \(I\) is prime whenever \(x+y\in I\) for \(x,y\in S\) implies \(x+y=0\) or \(x\in I\) or \(y\in I\). If \(I\subseteq S\) is a prime ideal then a mapping \(\varphi\colon S\to S\) such that \(\varphi(x)=0\) for \(x\in I\) and \(\varphi(x)=x\) for \(x\in S\setminus I\) is an endomorphism of \(S\). The paper presents an infinite commutative semigroup satisfying the identity \(x+x=y+y+y\) and containing exactly two prime ideals.
reviewed by: Václav Koubek
description: scientific article; zbMATH DE number 6054238