Injectivity of $$S$$ S -posets with respect to down closed regular monomorphisms

Abstract

No need to say that the study of injectivity with respect to different classes of monomorphisms is crucial in any category. In this paper, the notion of injectivity with respect to down closed embeddings in the category of S-posets, posets with a monotone action of a pomonoid S on them, is studied. We give a criterion, like the Baer condition for injectivity of modules, or Skornjakov criterion for injectivity of S-sets, for down closed injectivity. Also, we consider such injectivity for S itself, and its (po)ideals. Further, we investigate if this kind of injectivity is preserved or reflected by products, coproducts, direct sums, and order sums of S-posets. Finally, we define and study some kinds of weak injectivity with respect to down closed embeddings.

Order Vitally Dense Injectivity and Order Vitally Essentiality

“…We recall from [8] that a sub-S-poset A of S-poset B is called order dense, if for each b ∈ B there exists a ∈ A such that b ≤ a. Clearly any order dense sub-S-poset is order vitally dense.…”

“…Also, various kinds of closure operators are studied. In [8], Shahbaz investigates the down set closure operator in pos-S and studies injectivity with respect to the class of down closed embeddings (see [10] and [7]). We investigate the new closure operator on a commutative pomonoid, namely, an order vitally dense closure operator such that order vitally embedding emerges from this type of closure.…”

“…We study some properties of this closure operator and some categorical properties of this kind of embeddings. The injectivity with respect to different classes of monomorphisms has been studied in [6], [9], [8]. In [4], in the category of S-acts, the researchers study vital dense subact and vitally dense monomorphisms and injectivity with respect to this kind of monomorphisms.…”

Adjoint Relations Between the Category of Poset Acts and Some Other Categories

“…More precisely, as S-acts correspond to representations of monoids by transformations of sets, S-posets correspond to order preserving representations of pomonoids by order preserving transformations of posets. Preliminary work on properties of S-posets was done by Fakhruddin in the 1980s (see [8] and [9]), and was continued in recent papers [2,3,4,5,11,12,13,14]. In the present paper, actions of a pomonoid S on a set, S-acts as unary algebraic structures, are investigated as algebras in the category Pos.…”

Weak Factorization System for Actions of Po-monoids on Posets

“…[12]). A possibly empty sub S-poset A of an S-poset B is said to be down-closed in B if for each a ∈ A and b ∈ B with b ≤ a we have b ∈ A.…”