Sequentially Dense Essential Monomorphisms of Acts Over Semigroups

The class Md of sequentially dense monomorphisms were first defined and studied by Giuli, Ebrahimi, and Mahmoudi for projection algebras (acts over the monoid (N∞, min), of interest to computer scientists, as studied by Herrlich, Ehrig, and some others) and generalized to acts over arbitrary semigroups. Mdinjectivity has been shown by some of the above authors to be also useful in the study of ordinary injectivity of acts. Essentiality is an important notion closely related to injectivity. In this paper, we study essentiality with respect to sequentially dense monomorphisms of acts. We will show that the three different definitions of essentiality usually used in literature with respect to a subclass of monomorphisms are equivalent for the class of sequentially dense monomorphisms and, among other things, give some criterion to characterize and describe them explicitly. Also, we show the existence and the explicit description of a maximal such essential extension for any given act.