The flatness properties of \(S\)-poset \(A(I)\) and Rees factor \(S\)-posets
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 \(|S|=1\).
scientific article; zbMATH DE number 5377134