Proper Behaviour of Sequential Injectivity of Acts Over Semigroups
M. Ebrahimi, Mojgan Mahmoudi, Leila Shahbaz · Communications in Algebra · 2009
Banaschewski defines and gives sufficient conditions on a category 𝒜 and a subclass ℳ of its monomorphisms under which ℳ-injectivity properly behaves with respect to the notions such as ℳ-absolute retract, ℳ-essentialness, and the existence of ℳ-injective hulls. In this article, taking 𝒜 to be the category of acts over a semigroup S and ℳ d to be the class of sequentially dense monomorphisms, introduced and studied by Ebrahimi, Giuli, Mahmoudi, Moghaddasi, and Shahbaz, we study the behaviour of ℳ d -injectivity (also called sequentially dense injectivity). We show, among other things, that S 2 = S is a sufficient, but not necessary, condition for the behaviour of s-injectivity. Finally, we give an explicit description of the s-injective hulls of acts when they exist.