The interval number i(P) of a poset P is the smallest t such that P is a containment of sets that are unions of at most t real intervals. For the special poset Bn(k) consisting of the singletons and ksubsets of an n-element set, ordered by inclusion, i(Bn(k)) = min {k, n − k + 1} if |n/2 − k | ≥ n/2 − (n/2). For bipartite posets with n elements or n minimal elements, i(P) ≤  n lgn − lglgn  + ...

From the point of view of modal logic, coalgebraic logic over posets is the natural coalgebraic generalisation of positive modal logic. From the point of view of coalgebra, posets arise if one is interested in simulations as opposed to bisimulations. From a categorical point of view, one moves from ordinary categories to enriched categories. We show that the basic setup of coalgebraic logic ext...

As a generalization of countably C-approximating posets, the concept of countably QC-approximating posets is introduced. With the countably QC-approximating property, some characterizations of generalized completely distributive lattices and generalized countably approximating posets are given. The main results are as follows: (1) a complete lattice is generalized completely distributive if and...

The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops results of Raney on completely distributive lattices and of Hoofman on continuous Scott domains, and also generalizes Smyth’s “R-structures”. Various constructions on continuous poset...

It is a well known fact that a supersolvable lattice is ELoshellable. Hence a supersolvable lattice (resp., its Stanley-Reisner ring) is Cohen-Macaulay. We prove that if L is a supersolvable lattice such that all intervals have non-vanishing Mt~bius number, then for an arbitrary element x e L the poser L {x} is also Cohen-Macaulay. Posets with this property are called 2-Cohen-Macaulay posets. I...

We construct a family of posets, called signed Birkhoff posets, that may be viewed as signed analogs of distributive lattices. Our posets are generally not lattices, but they are shown to posses many combinatorial properties corresponding to well known properties of distributive lattices. They have the additional virtue of being face posets of regular cell decompositions of spheres. We give a c...

Lacking an explicit formula for the numbers T0(n) of all order relations (equivalently: T0 topologies) on n elements, those numbers have been explored only up to n = 13 (unlabeled posets) and n = 15 (labeled posets), respectively. In a new approach, we used an orderly algorithm to (i) generate each unlabeled poset on up to 14 elements and (ii) collect enough information about the posets on 13 e...

We characterize the linear inequalities satisfied by flag f -vectors of all finite bounded posets. We do the same for semipure posets. In particular, the closed convex cone generated by flag f -vectors of bounded posets of fixed rank is shown to be simplicial, and the closed cone generated by flag f -vectors of semipure posets of fixed rank is shown to be polyhedral. The extreme rays of both of...

the finite set of minimal elements is delivered following [1]. This is being achieved via adjacency and zeta matrix description of bipartite digraphs chains the representatives of graded posets. The bipartite digraphs elements of such chains amalgamate to form corresponding cover relation graded poset digraphs with corresponding adjacency matrices being amalgamated throughout natural join as sp...

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...