Affiliation:
1. Department of Mathematics , Tafresh University , 39518-79611 , Tafresh , Iran
Abstract
Abstract
For a class 𝓜 of monomorphisms of a category, mathematicians usually use different types of essentiality. Essentiality is an important notion closely related to injectivity. Banaschewski defines and gives sufficient conditions on a category 𝓐 and a subclass 𝓜 of its monomorphisms under which 𝓜-injectivity well-behaves with respect to the notions such as 𝓜-absolute retract and 𝓜-essentialness.
In this paper, 𝓐 is taken to be the category of acts over a semigroup S and 𝓜
sd
to be the class of strongly s-dense monomorphisms. We study essentiality with respect to strongly s-dense monomorphisms of acts. Depending on a class 𝓜 of morphisms of a category 𝓐, In some literatures, three different types of essentialness are considered. Each has its own benefits in regards with the behavior of 𝓜-injectivity. We will show that these three different definitions of essentiality with respect to the class of strongly s-dense monomorphisms are equivalent. Also, the existence and the explicit description of a strongly s-dense injective hull for any given act which is equivalent to the maximal such essential extension and minimal strongly s-dense injective extension with respect to strongly s-dense monomorphism is investigated. At last we conclude that strongly s-dense injectivity well behaves in the category Act-S.
Reference10 articles.
1. Banaschewski, B.: Injectivity and essential extensions in equational classes of algebras. Queens Papers in Pure and Appl. Math. 25, 1970, pp. 131–147.
2. Banaschewski, B.–-Nelson, E.: Equational compactness in equational classes of algebras, Algebra Universalis 2 (1972), 152–165.
3. Barzegar, H.: Strongly s-dense monomorphism, J. Hyperstructures 1(1) (2012), 14–26.
4. Barzegar, H.: Sequentially compactS-act, Journal of Algebraic Systems 5(2) (2017), 111–125.
5. Barzegar, H.–-Ebrahimi, M. M.–-Mahmoudi, M.: Essentiality and injectivity, Appl. Categ. Structures 18(1) (2010), 73–83.