Nets, useful topological tools, used to generalize certainconcepts that may only be general enough in the context of metricspaces. In this work we introduce this concept in an $S$-poset, aposet with an action of a posemigroup $S$ on it whichis a very useful structure in computer sciences and interestingfor mathematicians, and give the the concept of $S$-net. Using $S$-nets and itsconvergency we...

Let S-Set be the category of $S$-sets, sets together with the actions of a semigroup $S$ on them. And, let S-Pos be the category of $S$-posets, posets together with the actions compatible with the orders on them. In this paper we show that the category S-Pos is a radical extension of S-Set; that is there is a radical on the category S-Pos, the order desolator radical, whose torsion-free class i...

in (golchin a. and rezaei p., subpullbacks and flatness properties of s-posets. comm. algebra. 37: 1995-2007 (2009)) study was initiated of flatness properties of right -posets  over a pomonoid  that can be described by surjectivity of  corresponding to certain (sub)pullback diagrams and new properties such as  and  were discovered. in this article first of all we describe po-flatness propertie...

this work is devoted to give a charcaterization of a pomonoid $s$such that all cyclic $s$-posets are regular injective.

Two partially ordered monoids S and T are called Morita equivalent if the categories of right S-posets and right T -posets are Pos-equivalent as categories enriched over the category Pos of posets. We give a description of Pos-prodense biposets and prove Morita theorems I, II, and III for partially ordered monoids.

In 2001, S. Bulman-Fleming et al. initiated the study of three flatness properties (weakly kernel flat, principally weakly kernel flat, translation kernel flat) of right acts AS over a monoid S that can be described by means of when the functor AS ⊗ − preserves pullbacks. In this paper, we extend these results to S-posets and present equivalent descriptions of weakly kernel po-flat, principally...

‎‎‎In this paper‎, ‎the‎ notion of injectivity with respect to order dense 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 the order dense injectivity‎. ‎Also‎, ‎we consider such injectivit...

In this paper the notion of Rees short exact sequence for S-posets is introduced, and we investigate the conditions for which these sequences are left or right split. Unlike the case for S-acts, being right split does not imply left split. Furthermore, we present equivalent conditions of a right S-poset P for the functor Hom(P;-) to be exact.