‎‎‎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 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...

S. Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529–563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The co...

In every dense poset P every maximal antichain S may be partitioned into disjoint subsets S1 and S2 , such that the union of the upset of S1 with the downset of S2 yields the entire poset: U(S1) [ D(S2) = P . To nd a similar splitting of maximal antichains in posets is NP{hard in general.

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...

In this paper, we investigate po-purity using finitely presented S-posets, and give some equivalent conditions under which an S-poset is absolutely po-pure. We also introduce strongly finitely presented S-posets to characterize absolutely pure S-posets. Similar to the acts, every finitely presented cyclic S-posets is isomorphic to a factor S-poset of a pomonoid S by a finitely generated right con...

1. Introduction. Let A" be a partially ordered set (poset) with respect to a relation ^, and possessing least and greatest elements O and / respectively. Let us call a subset S of A" up-directed (down-directed) if and only if for all xES and y ES there exists zGS such that 3^x, z^y (z^x, z^y). Following McShane [2], we call a subset K of X Dedekind-closed if and only if whenever S is an up-dire...

‎In this paper we take $mathcal A$ to be the category {bf Pos-S}‎ ‎of $S$-posets‎, ‎for a posemigroup $S$‎, ${mathcal M}_{pd}$ to‎ ‎be the class of partially ordered sequantially-dense‎ ‎monomorphisms and study the categorical properties‎, ‎such as‎ ‎limits and colimits‎, ‎of this class‎. ‎These properties are usually‎ ‎needed to study the homological notions‎, ‎such as injectivity‎, ‎of‎ ‎$S$-...

In this paper X denotes a set. Let L be a lattice. Note that Poset(L) has l.u.b.’s and g.l.b.’s. Let L be an upper-bounded lattice. One can verify that Poset(L) is upper-bounded. Let L be a lower-bounded lattice. One can verify that Poset(L) is lower-bounded. Let L be a complete lattice. Note that Poset(L) is complete. Let X be a set. Then X is an order in X . Let X be a set. The functor 〈X ,⊆〉...

We investigate the poset 〈P(X) ∪ {∅},⊂〉, where X is a countable ultrahomogeneous partial order and P(X) the set of suborders of X isomorphic to X. For X different from (resp. equal to) a countable antichain the order types of maximal chains in 〈P(X) ∪ {∅},⊂〉 are characterized as the order types of compact (resp. compact and nowhere dense) sets of reals having the minimum non-isolated. 2000 Math...