\(\mathcal{M}\)-injectivity in the category Act-S

Abstract

Injectivity is one of the central notions in many branches of mathematics.  Different kinds of injectivity with respect to the class of all monomorphisms and with respect to some special subclasses of monomorphisms in the category Act-S of acts over a semigroup S have been studied before.  In this paper, we take the category Act-S of acts over a semigroup S, and \(\mathcal{M}\) as an arbitrary subclass of monomorphisms, and study some kinds of injectivity with respect to \(\mathcal{M}\).  Also, the behaviour of these notions of injectivity with respect to products, coproducts, and direct sums is studied.  As a result we give some characterizations of semigroups.