Homological properties of semigroups S and dense injectivity of S-acts

Actions of a semigroup on a set have always been a useful tool to study mathematical structures, and recently have captured the interest of some computer scientists, too. For this reason and because of its close relation to the category of sets, one can take the category of S-acts, for a semigroup S, as the universe of discourse to study mathematical notions in it. The sequentially dense monomorphisms of acts, which are also of interest to computer scientists, was first defined by Giuli, Ebrahimi and Mahmoudi, for projection algebras. Then this notion of sequentially dense monomorphisms was generalized to acts over an arbitrary semigroup and injectivity, which also may be said to be the most central notion in many branches of mathematics, with respect to them have been studied by Ebrahimi, Mahmoudi, and Moghaddasi. These encouraged the author to present some extensions of behaviour of this notion of injectivity.