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 sequential purity, for acts over semigroups (which is of some interest to computer scientists, too), 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 sequentially pure monomorphisms. These properties are usually needed to study the homological notions, such as injectivity, of acts.