Characterization of monoids by condition (P) of cyclic left acts (Q1328443)

Let \(S\) be a monoid. For a left \(S\)-act \(_ S A\), the following condition is denoted by (P): If \(sa = ta\), \(s,t \in S\), \(a \in A\), then there exist \(u \in S\), \(b \in A\) such that \(su = tu\) and \(a = ub\). A left \(S\)-act \(_ SA\) is called strongly flat if the functor \(-\otimes A\) preserves pullbacks. It is proved that: 1) all strongly flat left \(S\)- acts are projective generators in the category of all left \(S\)-acts if and only if \(S\) is a group or a group with \(0\) adjoined; 2) all strongly flat left \(S\)-acts are projective generators in the category of all left \(S\)-acts if and only if \(S\) is a group. Left reversible right PP monoids are described for which all cyclic flat left \(S\)-acts satisfy condition (P).