Periodic monoids over which all flat cyclic right acts satisfy condition (P) (Q677108)

Let \(S\) be a monoid. A right \(S\)-act \(A_S\) is flat (weakly flat) if the functor \(A\otimes-\) preserves monomorphisms (injections of left ideals of \(S\) into \(S\)). The following condition is denoted by (P): If \(as=a't\) for \(a,a'\in A\), \(s,t\in S\) then there exist \(a''\in A\) and \(u,v\in S\) such that \(a=a''u\), \(a'=a''v\), and \(us=vt\). It is proved that for a periodic monoid \(S\), the following conditions are equivalent: 1) \(S=G\dot\cup N\) where \(G\) is a group and either \(N=\emptyset\) or every element of \(N\) is right nil; 2) every weakly flat cyclic right \(S\)-act satisfies condition (P); 3) every flat cyclic right \(S\)-act satisfies condition (P). It is also shown that the implication \(1)\Rightarrow 2)\) holds for arbitrary monoids.