On monoids over which all strongly flat cyclic right acts are projective (Q1910649)

The author proves that for a monoid \(S\) all strongly flat cyclic right \(S\)-acts are projective if and only if every left collapsible submonoid \(P\) of \(S\) contains a left zero. A submonoid \(P\) is called left collapsible if for any \(p,q\in P\) there exists \(r\in P\) with \(rp=rq\). As a corollary one gets the equivalence of the following statements: (i) All strongly flat cyclic acts are free. (ii) All strongly flat cyclic acts are projective generators in \textbf{Act}-\(S\). (iii) The only left collapsible submonoid of \(S\) is the one-element submonoid.