On condition \((P')\).
Let \(S\) be a monoid. A right \(S\)-act \(A\) is said to satisfy `Condition \((P')\)' if for all \(a,a'\in A\), \(s,t,z\in S\), \(as=a't\) and \(sz=tz\) imply that there exist \(a''\in A\) and \(u,v\in S\) such that \(a=a''u\), \(a'=a''v\) and \(us=vt\). Monoids \(S\) are described over which all right acts in certain classes of acts satisfy Condition \((P')\). Among these classes are (a selection) 1) all right \(S\)-acts, 2) all injective right \(S\)-acts, 3) all torsion free right \(S\)-acts, 4) all cyclic right \(S\)-acts, 5) all right Rees factor \(S\)-acts. Conditions for some types of \(S\)-acts for satisfying Condition \((P')\) are found as well. For example, 1) \(\Theta_S\) satisfies Condition \((P')\) if and only if for every \(s,s',z\in S\), \(sz=s'z\) implies the existence of \(u,v\in S\) such that \(us=vs'\), and 2) the diagonal \(S\)-act \((S\times S)_S\) with \(|E(S)|\leq 2\) satisfies Condition \((P')\) if and only if \(S\) is right cancellative. Some results are also obtained in the classes of idempotent and regular monoids.
scientific article; zbMATH DE number 6176660