Categorical properties of sequentially dense monomorphisms of semigroup acts (Q717720)

Let \(S\) be a semigroup, and \textit{SubB} the lattice of all subacts of an \(S\)-act \(B\). A family \(C=(C_B)_{B\in \text{\textbf{Act-S}}}\), with \(C_B: SubB\to SubB\) is called a \textit{closure operator} on \textbf{Act-S} if it satisfies the following three conditions: 1) \(A\leq C_B(A)\), 2) \(A_1\leq A_2\) implies \(C_B(A_1) \leq C_B(A_2)\), 3) \(f(C_B(A))\leq C_C(f(A))\), for all morphisms \(f: B\to C\). A subact \(A\) of \(B\) is called \textit{C-closed} in \(B\) if \(C_B(A) = A\), and \textit{C-dense} in \(B\) if \(C_B(A) = B\). The closure operator defined as \(C^d_B(A) = \{ b\in B: bS\subseteq A\}\) for any subact \(A\) of \(B\) is called \textit{sequential closure operator} and denoted by \(C^d\). A \(C^d\)-dense subact is called \textit{sequentially dense} or \textit{s-dense}. An \(S\)-map \(f: A\to B\) is said to be \(s\)-dense if \(f(A)\) is an \(s\)-dense subact of \(B\). Some basic properties of the operator \(C^d\) and of the class \({\mathcal M}_d\) of sequentially dense monomorphisms are found. For example, it is proved that \({\mathcal M}_d\) is right and left cancellable, and is closed under limits and colimits.