Sequentially pure monomorphisms of acts over semigroups

Abstract

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.