C-essentialness and well-behavedness of C-injectivity in Act-S

Abstract

An important notion related to injectivity with respect to monomorphisms or any other class \(\mathcal{M}\) of morphisms in a category \(\mathcal{A}\) is essentialness.  In this paper, taking \(\mathcal{A}\) to be the category of right acts over a semigroup S, C to be an arbitrary closure operator in the category Act-S, and \(\mathcal{M}\)_d to be the class of C-dense monomorphisms resulting from a closure operator C, we study the properties of Md-essential monomorphisms and we show the existence of a maximal \(\mathcal{M}\)_d-essential extension for any given act.  Finally, the behavior of \(\mathcal{M}\)_d-injectivity in the sense that the three so called Well-behavedness propositions hold is studied.  We show that the idempotency and weak hereditariness of a closure operator C are sufficient, but not necessary, conditions for the well-behavedness of \(\mathcal{M}\)_d-injectivity.  The class of sequentially dense monomorphisms resulting from a special closure operator (sequential closure operator) and injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, Mahmoudi, Moghaddasi, and the author.  Some of these results generalize some of the results about the class of sequentially dense monomorphisms.