Flatness properties of acts: Some examples
type: Item
instance of: scholarly article
title: Flatness properties of acts: Some examples
zbMATH Open document ID: 0883.20029
DOI: 10.1007/PL00005918
author: Sydney Bulman-Fleming
author: Mati Kilp
published in: Semigroup Forum
publication date: 1998-01-25T00:00:00Z
review text: Let \(S\) be a monoid and let \(A_S\) be a right \(S\)-act. \(A_S\) is said to have property (P) if whenever \(au=a'v\), where \(a,a'\in A\) and \(u,v\in S\), there exist \(a_1\in A\) and \(s_1,t_1\in S\) such that \(a=a_1s_1\), \(a'=a_1t_1\), and \(s_1u=t_1v\). The authors present four examples answering in the negative some recently published conjectures about conditions when flat acts satisfy condition (P). Let us describe here one of these examples. Let \(S=\langle x,y\mid xy=x=yx\rangle\cup\{1\}\). Then \(S\) is a commutative monoid with exactly one idempotent, containing elements that are neither cancellative nor nil. Moreover, every weakly flat \(S\)-act has property (P). This answers the conjecture of Golchin and Renshaw that all (weakly) flat cyclic \(S\)-acts have property (P) if and only if \(S=C\cup N\), where \(C\) is the right cancellative submonoid of \(S\), \(N\) is either empty or consists of right nil elements of \(S\), and for every \(a\in C\), \(b\in N\), \(b\) belongs to \(Sab\).
Mathematics Subject Classification ID: 20M50
zbMATH DE Number: 1057422
zbMATH Keywords: flat right acts
zbMATH Keywords: flatness
zbMATH Keywords: commutative monoids
zbMATH Keywords: weakly flat acts
reviewed by: Peeter Normak
MaRDI profile type: Publication
full work available at URL: pl00005918
OpenAlex ID: W2071238676
description: scientific article; zbMATH DE number 1057422