Any notion of  purity  is normally defined in terms of
solvability of some set of equations. In this paper we first take
 this point of view to introduce a kind of purity, called
$s$-purity, for acts over semigroups (which is of some interest
to computer scientists), and then show that it is actually
equivalent to $C^p$-purity resulting from a closure operator.

The main objective of the paper is to study properties of the
category of all acts over a semigroup with respect to $C^p$-pure
monomorphisms. These properties are needed to study the
homological notions, such as injectivity, of acts. given
act.