Flatness properties of acts: Some examples (Q1365569)

Let \(S\) be a monoid and let \(A_S\) be a right \(S\)-act. \(A_S\) is said to have property (P) if whenever \(au=a'v\), where \(a,a'\in A\) and \(u,v\in S\), there exist \(a_1\in A\) and \(s_1,t_1\in S\) such that \(a=a_1s_1\), \(a'=a_1t_1\), and \(s_1u=t_1v\). The authors present four examples answering in the negative some recently published conjectures about conditions when flat acts satisfy condition (P). Let us describe here one of these examples. Let \(S=\langle x,y\mid xy=x=yx\rangle\cup\{1\}\). Then \(S\) is a commutative monoid with exactly one idempotent, containing elements that are neither cancellative nor nil. Moreover, every weakly flat \(S\)-act has property (P). This answers the conjecture of Golchin and Renshaw that all (weakly) flat cyclic \(S\)-acts have property (P) if and only if \(S=C\cup N\), where \(C\) is the right cancellative submonoid of \(S\), \(N\) is either empty or consists of right nil elements of \(S\), and for every \(a\in C\), \(b\in N\), \(b\) belongs to \(Sab\).