The flatness properties of \(S\)-poset \(A(I)\) and Rees factor \(S\)-posets
type: Item
instance of: scholarly article
title: The flatness properties of \(S\)-poset \(A(I)\) and Rees factor \(S\)-posets
zbMATH Open document ID: 1157.06008
author: Husheng Qiao
author: Fang Li
published in: Semigroup Forum
publication date: 2008-12-02T00:00:00Z
review text: The particular \(S\)-poset \(A(I)\) for a proper right ideal \(I\) of a partially ordered monoid \(S\) is studied: \(A(I)=[\{x,y\}\times(S-I)]\cup [\{z\}\times I]\) for \(x,y,z\notin S\), endowed with a certain right \(S\)-action and a certain partial order. Several flatness properties of \(A(I)\) are characterized, as (principally) weakly flat or (principally) weakly po-flat. Also, (po-) torsion freeness of \(A(I)\) is described. Finally, for partially ordered monoids \((S,\cdot, \leq)\) and proper convex right ideals \(I\) of \(S\), the Rees factor \(S\)-posets \(S/K\) are dealt with (note that \(S/K\) does not mean the usual Rees factor semigroup). In particular, those \(S/K\) are characterized which satisfy a certain condition \(P_\omega\). In the case where the identity of \(S\) is either minimal or maximal in \((S,\leq)\), the following is shown: (i) every right Rees factor \(S\)-poset of \(S\) is projective (strongly flat) if and only if \(|S|=1\) or \(S\) is a partially ordered group with 0 adjoined; (ii) every right Rees factor \(S\)-poset is free if and only if \(|S|=1\).
reviewed by: H. Mitsch
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 5377134
zbMATH Keywords: flatness property
zbMATH Keywords: \(S\)-poset \(A(I)\)
zbMATH Keywords: Rees factor \(S\)-posets
MaRDI profile type: Publication
full work available at URL: s00233-008-9083-3
OpenAlex ID: W2465083528
cites work: Flatness Properties of S-Posets
cites work: Lazard's Theorem for S ‐posets
cites work: The category of \(S\)-posets.
cites work: When all \(S\)-posets are principally weakly flat.
cites work: INDECOMPOSABLE, PROJECTIVE, AND FLATS-POSETS
cites work: Strongly Flat andPO-FlatS-Posets
DOI: 10.1007/S00233-008-9083-3
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description: scientific article; zbMATH DE number 5377134