The generalized Füredi conjecture holds for finite linear lattices

We say that a rank-unimodal poset P has rapidly decreasing rank numbers, or the RDR property, if above (resp. below) the largest ranks of P, the size of each level is at most half of the previous (resp. next) one. We show that a finite rank-unimodal, rank-symmetric, normalized matching, RDR poset of width w has a partition into w chains such that the sizes of the chains are one of two consecutive integers. In particular, there exists a partition of the linear lattices L_n(q) (subspaces of an n-dimensional vector space over a finite field, ordered by inclusion) into chains such that the number of chains is the width of L_n(q) and the sizes of the chains are one of two consecutive integers.     

On the representation of partially ordered sets

It is well known that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, ?) such that the supremum and the infimum of any finite setF ofP correspond, respectively to the union and intersection of the images of the elements ofF. Here necessary and sufficient conditions are given for similar isomophic representation of a poset where however the supremum and infimum of also infinite subsetsI correspond to the union and intersection of images of elements ofI.     

Irreducible posets with large height exist

The dimension of a poset (X, P) is the minimum number of linear extensions of P whose intersection is P. A poset is irreducible if the removal of any point lowers the dimension. If A is an antichain in X and X ? A ≠ Ø, then dim X ≤ 2 width ((X ? A) + 1. We construct examples to show that this inequality is best possible; these examples prove the existence of irreducible posets of arbitrarily large height. Although many infinite families of irreducible posets are known, no explicity constructed irreducible poset of height larger than four has been found.     

Unit Interval Orders of Open and Closed Intervals

A poset P = (X, ?) is a unit OC interval order if there exists a representation that assigns an open or closed real interval I(x) of unit length to each x ∈ P so that x ? y in P precisely when each point of I (x) is less than each point in I (y). In this paper we give a forbidden poset characterization of the class of unit OC interval orders and an efficient algorithm for recognizing the class. The algorithm takes a poset P as input and either produces a representation or returns a forbidden poset induced in P.     

Posets with Cover Graph of Pathwidth two have Bounded Dimension

Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant d such that if P is a poset with cover graph of P of pathwidth at most 2, then dim(P)=d. We answer this question in the affirmative by showing that d=17 is sufficient. We also show that if P is a poset containing the standard example S ₅ as a subposet, then the cover graph of P has treewidth at least 3.     

Dimension of the crown Skn

In 1941, Dushnik and Miller introduced the concept of the dimension of a poset (X, P) as the minimum number of linear extensions of P whose intersection is exactly P. Although Dilworth has given a formula for the dimension of distributive lattices, the general problem of determining the dimension of a poset is quite difficult. An equally difficult problem is to classify those posets which are dimension irreducible, i.e., those posets for which the removal of any point lowers the dimension. In this paper, we construct for each n≥3, k≥0, a poset, called a crown and denoted S^k_n, for which the dimension is given by the formula $\text{2?(n+k)}\text{(k+2)}$. Furthermore, for each t≥3, we show that there are infinitely many crowns which are irreducible and have dimension t. We then demonstrate a method of combining a collection of irreducible crowns to form an irreducible poset whose dimension is the sum of the crowns in the collection. Finally, we construct some infinite crowns possessing combinatorial properties similar to finite crowns.     

Forcing Posets with Large Dimension to Contain Large Standard Examples

The dimension of a poset P, denoted \(\dim (P)\), is the least positive integer d for which P is the intersection of d linear extensions of P. The maximum dimension of a poset P with \(|P|\le 2n+1\) is n, provided \(n\ge 2\), and this inequality is tight when P contains the standard example \(S_n\). However, there are posets with large dimension that do not contain the standard example \(S_2\). Moreover, for each fixed \(d\ge 2\), if P is a poset with \(|P|\le 2n+1\) and P does not contain the standard example \(S_d\), then \(\dim (P)=o(n)\). Also, for large n, there is a poset P with \(|P|=2n\) and \(\dim (P)\ge (1-o(1))n\) such that the largest d so that P contains the standard example \(S_d\) is o(n). In this paper, we will show that for every integer \(c\ge 1\), there is an integer \(f(c)=O(c^2)\) so that for large enough n, if P is a poset with \(|P|\le 2n+1\) and \(\dim (P)\ge n-c\), then P contains a standard example \(S_d\) with \(d\ge n-f(c)\). From below, we show that \(f(c)={\varOmega }(c^{4/3})\). On the other hand, we also prove an analogous result for fractional dimension, and in this setting f(c) is linear in c. Here the result is best possible up to the value of the multiplicative constant.     

Posets and VPG Graphs

We investigate the class of intersection graphs of paths on a grid (VPG graphs), and specifically the relationship between the bending number of a cocomparability graph and the poset dimension of its complement. We show that the bending number of a cocomparability graph G is at most the poset dimension of the complement of G minus one. Then, via Ramsey type arguments, we show our upper bound is best possible.     

On the greedy dimension of a partial order

This paper introduces a new concept of dimension for partially ordered sets. Dushnik and Miller in 1941 introduced the concept of dimension of a partial order P, as the minimum cardinality of a realizer, (i.e., a set of linear extensions of P whose intersection is P). Every poset has a greedy realizer (i.e., a realizer consisting of greedy linear extensions). We begin the study of the notion of greedy dimension of a poset and its relationship with the usual dimension by proving that equality holds for a wide class of posets including N-free posets, two-dimensional posets and distributive lattices.     

Markov Chains on Graded Posets

We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an up chain or down chain). The second type toggles between two adjacent rank levels (an up-and-down chain). We introduce two compatibility concepts between the up-directed transition probabilities (an up rule) and the down-directed (a down rule), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young’s lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.     

On the representation of partially ordered sets

It is well known that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, ⊆) such that the supremum and the infimum of any finite setF ofP correspond, respectively to the union and intersection of the images of the elements ofF. Here necessary and sufficient conditions are given for similar isomophic representation of a poset where however the supremum and infimum of also infinite subsetsI correspond to the union and intersection of images of elements ofI.     

Properties of the promotion Markov chain on linear extensions

The Tsetlin library is a very well-studied model for the way an arrangement of books on a library shelf evolves over time. One of the most interesting properties of this Markov chain is that its spectrum can be computed exactly and that the eigenvalues are linear in the transition probabilities. This result has been generalized in different ways by various people. In this work, we investigate one of the generalizations given by the extended promotion Markov chain on linear extensions of a poset P introduced by Ayyer et al. (J Algebr Comb 39(4):853–881, 2014). They showed that if the poset P is a rooted forest, the transition matrix of this Markov chain has eigenvalues that are linear in the transition probabilities and described their multiplicities. We show that the same property holds for a larger class of posets for which we also derive convergence to stationarity results.     

Dimension and Matchings in Comparability and Incomparability Graphs

We develop some new inequalities for the dimension of a finite poset. These inequalities are then used to bound dimension in terms of the maximum size of matchings. We prove that if the dimension of P is d and d=3, then there is a matching of size d in the comparability graph of P. There is no analogue of this result for cover graphs, as we show that there is a poset P of dimension d for which the maximum matching in the cover graph of P has size \(O(\log d)\). On the other hand, there is a dual result in which the role of chains and antichains is reversed, as we show that there is also a matching of size d in the incomparability graph of P. The proof of the result for comparability graphs has elements in common with Perles’ proof of Dilworth’s theorem. Either result has the following theorem of Hiraguchi as an immediate corollary: \(\dim (P)\le |P|/2\) when |P|=4.     

Retractions onto series-parallel posets

The poset retraction problem for a poset P is whether a given poset Q containing P as a subposet admits a retraction onto P, that is, whether there is a homomorphism from Q onto P which fixes every element of P. We study this problem for finite series-parallel posets P. We present equivalent combinatorial, algebraic, and topological charaterisations of posets for which the problem is tractable, and, for such a poset P, we describe posets admitting a retraction onto P.     

A generalization of the Dedekind–MacNeille completion

In this paper we introduce the notion of \(Z_{\delta }\)-continuity as a generalization of precontinuity, complete continuity and \(s_{2}\)-continuity, where Z is a subset selection. And for each poset P, a closure space \(Z^{c}_{\delta }(P)\) arises naturally. For any subset system Z, we define a new type of completion, called \(Z_{\delta }\)-completion, extending each poset P to a Z-complete poset. The main results are: (1) if a subset system Z is subset-hereditary, then \(cl_{Z}(\Psi (P))\), the Z-closure of all principal ideals \(\Psi (P)\) of poset P in \(Z^{c}_{\delta }(P)\), is a \(Z_{\delta }\)-completion of P and \(Z^{c}_{\delta }(P) \cong Z^{c}_{\delta }(cl_{Z}(\Psi (P)))\); (2) let Z be an HUL-system and P a \(Z_{\delta }\)-continuous poset, then the \(Z_{\delta }\)-completion of P is also \(Z_{\delta }\)-continuous, and a Z-complete poset L is a \(Z_{\delta }\)-completion of P iff P is an embedded \(Z_{\delta }\)-basis of L; (3) the Dedekind–MacNeille completion is a special case of the \(Z_{\delta }\)-completion.     

On-Line Dimension for Posets Excluding Two Long Incomparable Chains

For a positive integer k, let k?+?k denote the poset consisting of two disjoint k-element chains, with all points of one chain incomparable with all points of the other. Bosek, Krawczyk and Szczypka showed that for each k?≥?1, there exists a constant c _k so that First Fit will use at most $c_kw^2$ chains in partitioning a poset P of width at most w, provided the poset excludes k?+?k as a subposet. This result played a key role in the recent proof by Bosek and Krawczyk that O(w ^16logw ) chains are sufficient to partition on-line a poset of width w into chains. This result was the first improvement in Kierstead’s exponential bound: (5 ^w ???1)/4 in nearly 30 years. Subsequently, Joret and Milans improved the Bosek–Krawczyk–Szczypka bound for the performance of First Fit to 8(k???1)² w, which in turn yields the modest improvement to O(w ^14logw ) for the general on-line chain partitioning result. In this paper, we show that this class of posets admits a notion of on-line dimension. Specifically, we show that when k and w are positive integers, there exists an integer t?=?t(k,w) and an on-line algorithm that will construct an on-line realizer of size t for any poset P having width at most w, provided that the poset excludes k?+?k as a subposet.     

New Results on Global Rank Axioms of Poset Matroids

An excellent introduction to the topic of poset matroids is due to M. Barnabei, G. Nicoletti and L. Pezzoli. On the basis of their work, we have obtained the global rank axioms for poset matroids.In this paper, we study the special integral function f and obtain a new class of poset matroids from the old ones, and then we generalize this result according to the properties of f. Almost all of these results can be regarded as the application of global rank axioms for poset matroids. The main results in our paper have, indeed, investigated the restriction of the basis of the poset matroid, and we give them the corresponding geometric interpretation.     

Peckness of Edge Posets

For any graded poset P, we define a new graded poset, ??(P), whose elements are the edges in the Hasse diagram of P. For any group G acting on the boolean algebra B _n in a rank-preserving fashion we conjecture that ??(B _n /G) is Peck. We prove that the conjecture holds for “common cover transitive” actions. We give some infinite families of common cover transitive actions and show that the common cover transitive actions are closed under direct and semidirect products.     

A structure theorem for posets admitting a “strong” chain partition:: A generalization of a conjecture of Daykin and Daykin (with connections to probability correlation inequalities)

Suppose a finite poset P is partitioned into three non-empty chains so that, whenever p, q∈P lie in distinct chains and p<q, then every other element of P is either above p or below q.In 1985, the following conjecture was made by David Daykin and Jacqueline Daykin: such a poset may be decomposed into an ordinal sum of posets such that, for 1?i?n, one of the following occurs:

(1)

R_i is disjoint from one of the chains of the partition; or

(2)

if p, q∈R_i are in distinct chains, then they are incomparable.

The conjecture is related to a question of R. L. Graham's concerning probability correlation inequalities for linear extensions of finite posets.In 1996, a proof of the Daykin-Daykin conjecture was announced (by two other mathematicians), but their proof needs to be rectified.In this note, a generalization of the conjecture is proven that applies to finite or infinite posets partitioned into a (possibly infinite) number of chains with the same property. In particular, it is shown that a poset admits such a partition if and only if it is an ordinal sum of posets, each of which is either a width 2 poset or else a disjoint sum of chains. A forbidden subposet characterization of these partial orders is also obtained.     

The poset structures admitting the extended binary Golay code to be a perfect code

Brualdi et al. [Codes with a poset metric, Discrete Math. 147 (1995) 57-72] introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Golay code to be a 4-error-correcting perfect P-code. In this paper we classify all of the poset structures which admit the extended binary Golay code to be a 4-error-correcting perfect P-code, and show that there are no posets which admit the extended binary Golay code to be a 5-error-correcting perfect P-code.