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 (A,M), for a category A and a class M of monomorphisms in a category A. In this paper we take A to be the category Act-S of S-acts, for a semigroup S, and Msp to be the class of Csp I -pure monomorphisms and study some categorical and algebraic properties of this class concerning the closure operator Csp I .