We consider conditions which force a well-quasi-ordered poset (wqo) to be better-quasi-ordered (bqo). In particular we obtain that if a poset $P$ is wqo and the set $S_ω(P)$ of strictly increasing sequences of elements of $P$ is bqo under domination, then $P$ is bqo. As a consequence, we get the same conclusion if $S_ω (P)$ is replaced by $\mathcal J^1(P)$, the collection of non-principal ideals of $P$, or by $AM(P)$, the collection of maximal antichains of $P$ ordered by domination. It then follows that an interval order which is wqo is in fact bqo.
O. Einstein (2008) proved Bollobás-type theorems on intersecting families of ordered sets of finite sets and subspaces. Unfortunately, we report that the proof of a theorem on ordered sets of subspaces had a mistake. We prove two weaker variants.
In this article, we give a counter-example to Lemma 12 of the article "On Operations and Linear Extensions of Well Partially Ordered Sets" by Maciej Malicki and Aleksander Rutkowski.
An $α$-greedy balanced pair in an ordered set $P=(V,\leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of greedy linear extensions of $P$ that put $x$ before $y$ among all greedy linear extensions is in the real interval $[α, 1-α]$. We prove that every $N$-free ordered set which is not totally ordered has a $\frac{1}{2}$-greedy balanced pair.
Lattice path matroids form a subclass of transversal matroids and were introduced by Bonin, de Mier and Noy. Transversal matroids are not well-quasi-ordered, even when the branch-width is restricted. Though lattice path matroids are not well-quasi-ordered, we prove that lattice path matroids of bounded branch-width are well-quasi-ordered.
Discrete partially ordered sets can be turned into distance spaces in several ways. The distance functions may or may not satisfy the triangle inequality, and restriction of the distance to finite chains may or may not coincide with the natural, difference-of-height distance measured in a chain. For semilattices, a semimodularity condition ensures the good behavior of the distances considered. This condition is trivially satisfied by trees, and in lattices it coincides with the usual semimodularity property. For a large class of distance functions the triangle inequality is equivalent to semim…
We introduce several statistics on ordered partitions of sets, that is, set partitions where the blocks are permuted arbitrarily. The distribution of these statistics is closely related to the q-Stirling numbers of the second kind. Some of the statistics are generalizations of known statistics on set partitions, but others are entirely new. All the new ones are sums of two statistics, inspired by statistics on permutations, where one of the two statistics is based on a certain partial ordering of the blocks of a partition.
We consider a combinatorial question about searching for an unknown ideal $μ$ within a known poset $λ$. Elements of $λ$ may be queried for membership in $μ$, but at most $k$ positive query results are permitted. The goal is to find a search strategy which guarantees a solution in a minimal total number $m_k(λ)$ of queries. We provide tight bounds for $m_k(λ)$, and construct optimal search strategies for the case where $k=2$ and $λ$ is the product poset of totally ordered finite sets, one of which has cardinality not more than six.
Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled well-quasi-ordered, a notion stronger than that of $n$-well-quasi-order introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not $2$-well-quasi-ordered. This counterexa…
We describe one-dimensional central measures on numberings (tableaux) of ideals of partially ordered sets (posets). As the main example, we study the poset ${\Bbb Z}_+^d$ and the graph of its finite ideals, multidimensional Young tableaux; for $d=2$, it is the ordinary Young graph. The central measures are stratified by dimension; in the paper we give a complete description of the one-dimensional stratum and prove that every ergodic central measure is uniquely determined by its frequencies. The suggested method, in particular, gives the first purely combinatorial proof of E.~Thoma's theorem fo…
Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Gröbner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furt…
In this paper, nonzero component graphs and nonzero component union graphs of finite dimensional vector space are studied using the zero-divisor graph of specially constructed 0-1-distributive lattice and the zero-divisor graph of rings. Further, we define an equivalence relation on nonzero component graphs and nonzero component union graphs to deduce that these graphs are the graph join of zero-divisor graphs of Boolean algebras and complete graphs. In the last section, we characterize the perfect and chordal nonzero component graphs and nonzero component union graphs.
Let $Λ^{\ast}$ be the free monoid of (finite) words over a not necessarily finite alphabet $Λ$, which is equipped with some (partial) order. This ordering lifts to $Λ^{\ast}$, where it extends the divisibility ordering of words. The MacNeille completion of $Λ^{\ast}$ constitutes a complete lattice ordered monoid and is realized by the system of "closed" lower sets in $Λ^*$ (ordered by inclusion) or its isomorphic copy formed of the "closed" upper sets (ordered by reverse inclusion). Under some additional hypothesis on $Λ$, one can easily identify the closed lower sets as the finitely generated…
We prove that there are arbitrarily large indecomposable ordered sets T with a 2-chain C such that the smallest indecomposable proper superset U of C in T is T itself. Subsequently, we characterize all such indecomposable ordered sets T and 2-chains C. We also prove the same type of result for 2-antichains.
In this survey we present a generalization of the notion of metric space and some applications to discrete structures as graphs, ordered sets and transition systems. Results in that direction started in the middle eighties based on the impulse given by Quilliot (1983). Graphs and ordered sets were considered as kind of metric spaces, where - instead of real numbers - the values of the distance functions $d$ belong to an ordered semigroup equipped with an involution. In this frame, maps preserving graphs or posets are exactly the nonexpansive mappings (that is the maps $f$ such that $d(f(x),f(y…
We define a variation of Stirling permutations, called quasi-Stirling permutations, to be permutations on the multiset $\{1,1,2,2,\ldots, n,n\}$ that avoid the patterns 1212 and 2121. Their study is motivated by a known relationship between Stirling permutations and increasing ordered rooted labeled trees. We construct a bijection between quasi-Stirling permutations and the set of ordered rooted labeled trees and investigate pattern avoidance for these permutations.
