A closure operator versus purity

‎Any notion of purity is normally defined in terms of‎ ‎solvability of some set of equations‎. ‎To study mathematical notions‎, ‎such as injectivity‎, ‎tensor products‎, ‎flatness‎, ‎one needs to have some categorical and‎ ‎algebraic information about the pair (${mathcal A}$,${mathcal M}$)‎, ‎for a category $mathcal A$‎ ‎and a class $mathcal M$ of monomorphisms in a category $mathcal A$‎. ‎In this paper we take $mathcal A$ to be the category {bf Act-S}‎ ‎of $S$-acts‎, ‎for a semigroup $S$‎, ‎and ${mathcal M}_{sp}$ to be‎ ‎the class of $C_I^{sp}$-pure monomorphisms and study some‎ ‎categorical and algebraic properties of this class concerning the closure operator $C_I^{sp}$‎.