Essentiality and injectivity relative to sequential purity of‎ ‎acts

Abstract

For a classMof monomorphisms of a category, mathematicians consider
different types of essentiality, depending onM. In this paper, considering the category
of acts over a semigroup, we first briefly study the classMp of a certain kind of
pure monomorphisms, in a manner borrowed from V. Gould, to be called sequentially
pure. Then, we study in detail three kinds of essentiality with respect to this class, and
give some useful criteria to get (internal) characterizations (in terms of elements) for
essentialities. Finally, the relations between injectivity, essentiality, retractness, and
injective hulls, all with respect to the class of sequentially pure monomorphisms, are
investigated.