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Strongly s-dense injective hull and Banaschewski’s theorems for acts

Hasan Barzegar
From the journal Mathematica Slovaca

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 𝓜sd 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.

MSC 2010: 08A60; 18A20; 20M30; 20M50

  1. Communicated by Aleš Pultr

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Received: 2018-10-07
Accepted: 2019-10-18
Published Online: 2020-03-10
Published in Print: 2020-04-28

© 2020 Mathematical Institute Slovak Academy of Sciences