On \(P\)-regularity of acts. (Q2890848)

Let \(S\) be a monoid. A right \(S\)-act \(A\) is called `\(P\)-regular' if all its cyclic subacts satisfy the following condition (P): \(as=a's'\) implies the existence of \(a''\in A\), \(u,v\in S\) such that \(a=a''u\), \(a'=a''v\) and \(us=vs'\). Conditions for \(S\)-acts from different classes of acts to be \(P\)-regular are found. For Rees factor acts the opposite task is solved as well: conditions are found under which all \(P\)-regular right Rees factor acts have certain properties.