\(C\)-dense injectivity in Act-\(S\). (Q2911878)

Let \(S\) be a semigroup and \(\mathbf{Act}\)-\(S\) the category of all right \(S\)-acts and \(S\)-maps between them. For \(B\in\mathbf{Act}\)-\(S\), \(\text{Sub\,}B\) denotes the lattice of all subacts of \(B\). A family \(C=(C_B)_{B\in\mathbf{Act}\text{-}S}\) with \(C_B\colon\text{Sub\,}B\to\text{Sub\,}B\) is called a `closure operator' on \(\mathbf{Act}\)-\(S\) if it satisfies the following conditions: 1) \(A\leq C_B(A)\), 2) \(A_1\leq A_2\) implies \(C_B(A_1)\leq C_B(A_2)\), 3) \(f(C_B(A))\leq C_D(f(A))\) for all morphisms \(f\colon B\to D\). An \(S\)-act \(A\subseteq B\) is called `\(C\)-dense' in \(B\) if \(C_B(A)=B\) and an \(S\)-map \(f\colon A\to B\) is called \(C\)-dense if \(f(A)\) is \(C\)-dense in \(B\). An \(S\)-act \(A\) is called `\(C\)-dense injective' or `\(C\)-injective' if it is injective with respect to \(C\)-dense monomorphisms. Similarly, weakly \(C\)-injective, \(FC\)-injective, \(PC\)-injective, finitely \(C\)-injective and ideal \(C\)-injective \(S\)-acts are defined.NEWLINENEWLINE It turned out that the properties of \(C\)-injectivity are to a great extent similar to those of general injectivity. For example, an \(S\)-act is \(C\)-injective if and only if it is \(C\)-absolute retract. -- Some categorical properties of \(C\)-dense monomorphisms are established as well.