Flatness properties of monocyclic acts (Q1126386)

Let \(S\) be a monoid, \(\rho(s,t)\) is the smallest right congruence on \(S\) identifying the elements \(s\) and \(t\) of \(S\). Then \(S/\rho(s,t)\) is a right \(S\)-act and such acts are called monocyclic. Results of the following kind are contained in the paper. All projective monocyclic right \(S\)-acts are free if and only if all projective right \(S\)-acts are free, i.e. if and only if \(e{\mathcal D}1\) for all \(e^2=e\in S\). All possible combinations of these types or statements are presented where ``projective'' or ``free'' are replaced by ``strongly flat'', ``\((P)\)'', ``flat'', ``weakly flat'', ``principally weakly flat'', ``torsion free'' and, at the first place in the above statement, also ``all''. The possible 36 results are either proved or cited in the paper.