Flat acts that satisfy condition (P) (Q1300605)

The paper discusses monoids over which all acts satisfy certain flatness conditions. For example, it is proved that if a monoid \(S\) is such that all flat cyclic \(S\)-acts have property (P), then \(S\) is a disjoint union of a group, a right nil set and a regular-free set (a subset \(X\) of a monoid is called regular-free if it does not contain eventually regular elements; an element \(s\in X\) is called eventually regular if there exists a natural number \(n\) such that \(x^n\) is regular). This structure is used in simplifying some of the main results in the literature.