Strongly $s$-dense injective hull and Banaschewski's theorems for acts

Year, pages: 2020, 251 - 258

For a class M $\mathcal M$ 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 $\mathcal A$ and a subclass $\mathcal M$ of its monomorphisms under which $\mathcal M$-injectivity well-behaves with respect to the notions such as $\mathcal M$-absolute retract and $\mathcal M$-es sen tial ness. In this paper, $\mathcal A$ is taken to be the category of acts over a semigroup $S$ and ${\mathcal M}_sd$ 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 $\mathcal M$ of morphisms of a category ${\mathcal A}$, In some literatures, three different types of essentialness are considered. Each has its own benefits in regards with the behavior of $\mathcal M$-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.

Barzegar, H. 2020. Strongly $s$-dense injective hull and Banaschewski's theorems for acts. In Mathematica Slovaca, vol. 70, no.2, pp. 251-258. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0348

Barzegar, H. (2020). Strongly $s$-dense injective hull and Banaschewski's theorems for acts. Mathematica Slovaca, 70(2), 251-258. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0348

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava