Down closed injectivity and essentialness

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 [Formula: see text]-posets, posets with an action of a pomonoid [Formula: see text] on them. They gave a criterion for dc-injectivity and studied such injectivity for [Formula: see text] itself, and for its poideals. In this paper, we give results about dc-injectivity of [Formula: see text]-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 [Formula: see text]-injectivity for [Formula: see text]-posets, where [Formula: see text] and [Formula: see text] is externally adjoined to the posemigroup [Formula: see text], we find some classes of pomonoids such that for [Formula: see text]-posets over them the Baer Criterion holds. Further, several kinds of essentiality of down closed monomorphisms of [Formula: see text]-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 [Formula: see text]-posets for some classes of pomonoids [Formula: see text].

Injective hulls of many-sorted ordered algebras

This paper is devoted to the study of injectivity for ordered universal algebras. We first characterize injectives in the category $\begin{array}{} \displaystyle {\mathsf{OAL}_{{\it\Sigma}}^{\leqslant}} \end{array}$ of ordered Σ-algebras with lax morphisms as sup-Σ-algebras. Second, we show that every ordered Σ-algebra has an σ⩽-injective hull, and give its concrete form.