Forbidden subposet problems in the grid

For posets P and Q, extremal and saturation problems about weak and strong P-free subposets of Q have been studied mostly in the case Q is the Boolean poset $Q_n$, the poset of all subsets of an n-element set ordered by inclusion. In this paper, we study some instances of the problem with Q being the grid, and its connections to the Boolean case and to the forbidden submatrix problem.     

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.     

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.     

On Beck’s coloring of posets

We study Beck-like coloring of partially ordered sets (posets) with a least element 0. To any poset P with 0 we assign a graph (called a zero-divisor graph) whose vertices are labelled by the elements of P with two vertices x,y adjacent if 0 is the only element lying below x and y. We prove that for such graphs, the chromatic number and the clique number coincide. Also, we give a condition under which posets are not finitely colorable.     

A self paired Hopf algebra on double posets and a Littlewood-Richardson rule

Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood-Richardson rule.     

Linear Discrepancy of Chain Products and Posets with Bounded Degree

The linear discrepancy of a poset P, denoted ld(P), is the minimum, over all linear extensions L, of the maximum distance in L between two elements incomparable in P. With r denoting the maximum vertex degree in the incomparability graph of P, we prove that \({\rm ld}(P)\le \left\lfloor (3r-1)/2 \right\rfloor\) when P has width 2. Tanenbaum, Trenk, and Fishburn asked whether this upper bound holds for all posets. We give a negative answer using a randomized construction of bipartite posets whose linear discrepancy is asymptotic to the trivial upper bound 2r???1. For products of chains, we give alternative proofs of results obtained independently elsewhere.     

Upper Maximal Graphs of Posets

We introduce and study a class of simple graphs, the upper-maximal graphs (UM-graphs), associated to finite posets. The vertices of the UM-graph of a given poset P are the elements of P, and edges are formed by those vertices x and y whenever any maximal element of P that is greater than x is also greater than y or vise versa. We show that the class of UM-graphs constitutes a subclass of comparability graphs. We further provide a characterization of chordal UM-graphs, and compare UM-graphs with known bound graphs of posets.     

Minimizing the jump number for partially ordered sets: A graph-theoretic approach

The purpose of this paper is to present a graph-theoretic approach to the jump number problem for N-free posets which is based on the observation that the Hasse diagram of an N-free poset is a line digraph. Therefore, to every N-free poset P we can assign another digraph which is the root digraph of the Hasse diagram of P. Using this representation we show that the jump number of an N-free poset is equal to the cyclomatic number of its root digraph and can be found (without producing any linear extension) by an algorithm which tests if a given poset is N-free. Moreover, we demonstrate that there exists a correspondence between optimal linear extensions of an N-free poset and spanning branchings of its root digraph. We provide also another proof of the fact that optimal linear extensions of N-free posets are exactly greedy linear extensions. In conclusion, we discuss some possible generalizations of these results to arbitrary posets.     

On double bound graphs and forbidden subposets

For a poset P=(X,≤_P), the double bound graph (DB-graph) of P is the graph DB(P)=(X,E_DB(P)), where xy∈E_DB(P) if and only if x≠y and there exist n,m∈X such that n≤_Px,y≤_Pm. We obtain that for a subposet Q of a poset P,Q is an (n, m)-subposet of P if and only if DB(Q) is an induced subgraph DB(P). Using this result, we show some characterizations of split double bound graphs, threshold double bound graphs and difference double bound graphs in terms of (n, m)-subposets and double canonical posets.     

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.     

3-Interval irreducible partially ordered sets

In this paper we discuss the characterization problem for posets of interval dimension at most 2. We compile the minimal list of forbidden posets for interval dimension 2. Members of this list are called 3-interval irreducible posets. The problem is related to a series of characterization problems which have been solved earlier. These are: The characterization of planar lattices, due to Kelly and Rival [5], the characterization of posets of dimension at most 2 (3-irreducible posets) which has been obtained independently by Trotter and Moore [8] and by Kelly [4] and the characterization of bipartite 3-interval irreducible posets due to Trotter [9].We show that every 3-interval irreducible poset is a reduced partial stack of some bipartite 3-interval irreducible poset. Moreover, we succeed in classifying the 3-interval irreducible partial stacks of most of the bipartite 3-interval irreducible posets. Our arguments depend on a transformationP B(P), such that IdimP=dimB(P). This transformation has been introduced in [2].Supported by the DFG under grant FE 340/2–1.     

Boolean Lattices: Ramsey Properties and Embeddings

A subposet Q ^′ of a poset Q is a copy of a poset P if there is a bijection f between elements of P and Q ^′ such that x ≤ y in P iff f(x) ≤ f(y) in Q. For posets P, P ^′, let the poset Ramsey number R(P, P ^′) be the smallest N such that no matter how the elements of the Boolean lattice Q _N are colored red and blue, there is a copy of P with all red elements or a copy of P ^′ with all blue elements. We provide some general bounds on R(P, P ^′) and focus on the situation when P and P ^′ are both Boolean lattices. In addition, we give asymptotically tight bounds for the number of copies of Q _n in Q _N and for a multicolor version of a poset Ramsey number.     

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.     

Cover-preserving order embeddings into Boolean lattices

It is not known which finite graphs occur as induced subgraphs of a hypercube. This is relevant in the theory of parallel computing. The ordered version of the problem is: Which finite posets P occur as cover-preserving subposets of a Boolean lattice? Our main Theorem gives (for 0,1-posets) a necessary and sufficient condition, which involves the chromatic number of a graph associated to P. It is applied respectively to upper balanced, meet extremal, meet semidistributive, and semidistributive lattices P. More specifically, we consider isometric embeddings of posets into Boolean lattices. In particular, answering a question of Ivan Rival to the positive, a nontrivial invariant for the covering graph of a poset is found.     

A note on random k-dimensional posets

In 2009, Janson [Poset limits and exchangeable random posets, Institut Mittag-Leffler preprint, 36pp, arXiv:0902.0306] extended the recent theory of graph limits to posets, defining convergence for poset sequences and proving that every such sequence has a limit object. In this paper, we focus on k-dimensional poset sequences. This restriction leads to shorter proofs and to a more intuitive limit object. As before, the limit object can be used as a model for random posets, which generalizes the well known random k-dimensional poset model. This investigation also leads to a definition of quasirandomness for k-dimensional posets, which can be captured by a natural distance that measures the discrepancy of a k-dimensional poset.     

Classification of the factorial functions of Eulerian binomial and Sheffer posets

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or ²n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n)=n! has Sheffer factorial function D(n) identical to that of the infinite Boolean algebra, the infinite Boolean algebra with two new coatoms inserted, or the infinite cubical poset. Moreover, we are able to classify the Sheffer factorial functions of Eulerian Sheffer posets with binomial factorial function B(n)=²n−1 as the doubling of an upside-down tree with ranks 1 and 2 modified. When we impose the further condition that a given Eulerian binomial or Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite Boolean algebra BX or the infinite cubical lattice . We also include several poset constructions that have the same factorial functions as the infinite cubical poset, demonstrating that classifying Eulerian Sheffer posets is a difficult problem.     

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.     

Harmonics on posets

Given a finite ranked poset P, for each rank of P a space of complex valued functions on P called harmonics is defined. If the automorphism group G of P is sufficiently rich, these harmonic spaces yield irreducible representations of G. A decomposition theorem, which is analogous to the decomposition theorem for spherical harmonics, is stated. It is also shown that P can always be decomposed into posets whose principal harmonics are orthogonal polynomials. Classical examples are given.     

Embedding finite posets in cubes

In this paper we define the n-cube Q_n as the poset obtained by taking the cartesian product of n chains each consisting of two points. For a finite poset X, we then define dim₂X as the smallest positive integer n such that X can be embedded as a subposet of Q_n. For any poset X we then have log₂ |X| ? dim₂X ? |X|. For the distributive lattice $\text{L =}\text{2}{}^{\mathrm{X}}$, dim₂L = |X| and for the crown S^k_n, dim₂ (S^k_n) = n + k. For each k ? 2, there exist positive constants c₁ and c₂ so that for the poset X consisting of all one element and k-element subsets of an n-element set, the inequality c₁ log₂n < dim₂(X) < c₂ log₂n holds for all n with k < n. A poset is called Q-critical if dim₂ (X ? x) < dim₂(X) for every x ? X. We define a join operation ⊕ on posets under which the collection Q of all Q-critical posets which are not chains forms a semigroup in which unique factorization holds. We then completely determine the subcollection $\text{M}$ ? Q consisting of all posets X for which dim₂ (X) = |X|.