Quasi-exact sequences of \(S\)-acts (Q2272624)

Let \(\rho\) be a congruence on an \(S\)-act \(C_S\). A sequence \(B_S\xrightarrow{f} A_S\xrightarrow{g} C_S\) of \(S\)-acts is called \(\rho\)-\textit{exact} (or \textit{quasi-exact}) at \(A_S\) if (\(\operatorname{Im} f\times \operatorname{Im} f)\cup\Delta_{A_S} = \{(a,b)\in A_S\times A_S\mid (g(a), g(b))\in\rho\}\). If \(f\) is a monomorphism and \(g\) an epimorphism in a \(\rho\)-exact sequence, then the sequence is called a \textit{short \(\rho\)-exact sequence} (or a \textit{short quasi-exact sequence}). It is proved that if \(B_S\) and \(C_S\) in a short \(\rho\)-exact sequence are principally weakly flat (satisfy condition \((E)\), respectively), then so is \(A_S\). Under some additional conditions, some results on \(\rho\)-exact sequences are obtained. For example, if \(f\) is a monomorphism in a \(\rho\)-exact sequence and \(B_S\) and \(C_S\) are regular \(S\)-acts, then so is \(A_S\). A generalization of the short five lemma and some related results on commutative diagrams of \(S\)-acts are proven.