Abstract
To study mathematical notions, such as injectivity with respect to the class [Formula: see text] of (mono)morphisms in a category [Formula: see text], one needs to have some categorical and algebraic information about the pair ([Formula: see text], [Formula: see text]). In this paper, we take [Formula: see text] to be the category Act-S of acts over a semigroup S, C to be an arbitrary closure operator in the category Act-S, and [Formula: see text] to be the class of C-dense monomorphisms resulting from a closure operator C and first study some categorical properties of the pair (Act-S, [Formula: see text]). Then injectivity with respect to the class of C-dense monomorphisms is studied. 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 and the author. Some of these results generalize some of the results about the class of sequentially dense monomorphisms.
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