The paper considers a generalization of the standard completion of a partially ordered set through the collection of all its lower sets.

We continue the study of partially ordered generalized patterns (POGPs) considered in [9] for permutations and in [10] for words. We deal with segmented POGPs (SPOGPs). We state some general results and treat a number of patterns of length 4. We prove a result from [8] in a much simpler way and also establish a connection between SPOGPs and walks on lattice points starting from the origin and r...

We review selected known results on partially ordered patterns (POPs) that include co-unimodal, multiand shuffle patterns, peaks and valleys ((modified) maxima and minima) in permutations, the Horse permutations and others. We provide several (new) results on a class of POPs built on an arbitrary flat poset, obtaining, as corollaries, the bivariate generating function for the distribution of pe...

We generalize the classical notion of majorization in Rn to a majorization order for functions defined on a partially ordered set P . In this generalization we use inequalities for partial sums associated with ideals in P . Basic properties are established, including connections to classical majorization. Moreover, we investigate transfers (given by doubly stochastic matrices), complexity issue...

We introduce the problem of compressing partially ordered strings: given string σ ∈ Σ* and a binary independence relation I over Σ how can we compactly represent an input if the decompressor is allowed to reconstruct any string that can be obtained from σ by repeatedly swapping adjacent independent symbols? Such partially ordered strings are also known as Mazurkiewicz traces, and naturally mode...

Let be a partial order on a countable set V such that there exists a minimum element a ∈ V and {u ∈ V : u v} is finite for each v ∈ V . An item of information is to be broadcast to the elements of V according to the policy in the following sense: a is initially informed and broadcasting to v 6= a begins as soon as all elements u ≺ v are informed; the time required to broadcast the information i...

We continue the study of partially ordered generalized patterns (POGPs) considered in [9] for permutations and in [10] for words. We deal with segmented POGPs (SPOGPs). We state some general results and treat a number of patterns of length 4. We prove a result from [8] in a much simpler way and also establish a connection between SPOGPs and walks on lattice points starting from the origin and r...

Fuchs [6] called a partially-ordered integral domain, say D, division closed if it has the property that whenever a > 0 and ab > 0, then b > 0. He showed that if D is a lattice-ordered division closed field, then D is totally ordered. In fact, it is known that for a lattice-ordered division ring, the following three conditions are equivalent: a) squares are positive, b) the order is total, and ...

We employ partially ordered sets to describe the stratification of a social system, using rank to define the strata. We present a simple method of computing the matrix corresponding to the Hasse diagram and prove its correctness. This methodology is applied to analyze the hierarchy of countries that have won at least one Olympic medal. Four different definitions of dominance are given, leading ...