Classification of monoids by injectivities. I: C-injectivity. (Q2480772)

For a monoid \(S\), a right \(S\)-act \(A_S\) is called `C-injective' if for any cyclic \(S\)-act \(M_S\), any homomorphism \(f\colon M_S\to A_S\) can be extended to any right \(S\)-act \(N_S\supseteq M_S\). It is proved that all right ideals of \(S\) (or, equivalently, all principal right ideals of \(S\)) are C-injective if and only if \(S\) is a regular self-injective monoid. All \(S\)-acts (equivalently, all cyclic \(S\)-acts) are C-injective if and only if \(S\) has a left zero and for every right ideal \(K\) of \(S\), right congruences \(\mu,\lambda\) on \(S\), and every homomorphism \(f\in\Hom(\overline K_\mu,S/\lambda\)), there exists an element \(q\) of \(S\) such that \(f([m]_\mu)=[q]_\lambda m\) for each \([m]_\mu\in\overline K_\mu\), and \(s{\mathcal R}(K,\mu,\lambda,q)t\), \(s,t\in S\), implies that \((qs)\lambda(qt)\), where \((K,\mu,\lambda,q)\) is a certain right congruence on \(S\).