Injectivity is one of the useful notions in algebra, as well as in many other branches of mathematics, and the study of injectivity with respect to different classes of monomorphisms is crucial in many categories. Also, essentiality is an important notion closely related to injectivity. Down closed monomorphisms and injectivity with respect to these monomorphisms, so-called dc-injectivity, were first introduced and studied by the authors for S-posets, posets with an action of a pomonoid S on them. They gave a criterion for dc-injectivity and studied such injectivity for S itself, and for its poideals. In this paper, we give results about dc-injectivity of S-posets, also we find some homological characterization of pomonoids and pogroups by dc-injectivity. In particular, we give a characterization of pomonoids over which dc-injectivity is equivalent to having a zero top element. Also, introducing the notion of T-injectivity for S-posets, where S = T∪̇{1} and 1 is externally adjoined to the posemigroup T, we find some classes of pomonoids such that for S-posets over them the Baer Criterion holds. Further, several kinds of essentiality of down closed monomorphisms of S-posets, and their relations with each other and with dc-injectivity is studied. It is proved that although these essential extensions are not necessarily equivalent, they behave almost equivalently with respect to dc-injectivity. Finally, we give an explicit description of dc-injective hulls of S-posets for some classes of pomonoids S.

中文翻译：

---

向下封闭的注入性和本质性

内射性是代数以及许多其他数学分支中的有用概念之一，关于不同类别的单态性的内射性研究在许多类别中都是至关重要的。此外，本质是与内射性密切相关的一个重要概念。作者首先介绍和研究了关于这些单态的下闭单态和内射性，即所谓的直流内射性。小号-posets，带有 pomonoid 动作的posets小号在他们。他们给出了直流内射性的标准，并研究了这种内射性小号本身，以及它的理想。在本文中，我们给出了关于 dc-injectivity 的结果小号-posets，我们还发现了一些通过直流内射性对 pomonoids 和 pogroups 的同调表征。特别是，我们给出了 pomonoids 的特征，其中 dc-injectivity 等效于具有零顶部元素。此外，引入的概念吨- 内射性小号-Posets，在哪里小号 = 吨∪̇{1}和1在外部连接到波塞米群吨, 我们发现一些类 pomonoids 使得小号- 对他们提出了贝尔标准的要求。进一步，下闭单态的几种本质小号-posets，以及它们之间的关系以及与 dc-injectivity 的关系。事实证明，虽然这些基本扩展不一定是等价的，但它们在直流内射方面的表现几乎是等价的。最后，我们给出了 dc-injective hulls 的明确描述小号- 某些类 pomonoids 的设置小号.