Minimum cutsets for an element of a boolean lattice

An informative new proof is given for the theorem of Nowakowski that determines for all n and k the minimum size of a cutset for an element A with |A|=k of the Boolean algebra B _n of all subsets of {1,...,n}, ordered by inclusion. An inequality is obtained for cutsets for A that is reminiscent of Lubell's inequality for antichains in B _n. A new result that is provided by this approach is a list of all minimum cutsets for A.Research supported in part by NSF Grant DMS 87-01475.Research supported in part by NSF Grant DMS 86-06225 and Air Force OSR-86-0076.     

Projective mappings between projective lattice geometries

The concept of projective lattice geometry generalizes the classical synthetic concept of projective geometry, including projective geometry of modules.In this article we introduce and investigate certain structure preserving mappings between projective lattice geometries. Examples of these so-calledprojective mappings are given by isomorphisms and projections; furthermore all linear mappings between modules induce projective mappings between the corresponding projective geometries.     

Solution to a problem of C. D. Godsil regarding bipartite graphs with unique perfect matching

We give the solution to the following question of C. D. Godsil[2]: Among the bipartite graphsG with a unique perfect matching and such that a bipartite graph obtains when the edges of the matching are contracted, characterize those having the property thatG ⁺G, whereG ⁺ is the bipartite multigraph whose adjacency matrix,B ⁺, is diagonally similar to the inverse of the adjacency matrix ofG put in lower-triangular form. The characterization is thatG must be obtainable from a bipartite graph by adding, to each vertex, a neighbor of degree one. Our approach relies on the association of a directed graph to each pair (G, M) of a bipartite graphG and a perfect matchingM ofG.     

Explicit matchings in the middle levels of the Boolean lattice

New classes of explicit matchings for the bipartite graph (k) consisting of the middle two levels of the Boolean lattice on 2k+1 elements are constructed and counted. This research is part of an ongoing effort to show that (k) is Hamiltonian.Supported by Office of Naval Research contract N00014-85K-0494.Supported by National Science Foundation grant DMS-8041281.     

Maximum antichains in the product of chains

Let P be the poset k ₁ × ... × k _n , which is a product of chains, where n1 and k ₁ ... k _n 2. Let . P is known to have the Sperner property, which means that its maximum ranks are maximum antichains. Here we prove that its maximum ranks are its only maximum antichains if and only if either n=1 or M1. This is a generalization of a classical result, Sperner's Theorem, which is the case k ₁= ... =k _n =2. We also determine the number and location of the maximum ranks of P.Research supported in part by the National Science Foundation 10/25/83.     

General lexicographic shellability and orbit arrangements

We introduce a new poset property which we call EC-shellability. It is more general than the more established concept of EL-shellability, but it still implies shellability. Because of Theorem 3.10, EC-shellability is entitled to be called general lexicographic shellability. As an application of our new concept, we prove that intersection lattices Π_λ of orbit arrangementsA _λ are EC-shellable for a very large class of partitions λ. This allows us to compute the topology of the link and the complement for these arrangements. In particular, for this class of λs, we are able to settle a conjecture of Björner [B94, Conjecture 13.3.2], stating that the cohomology groups of the complement of the orbit arrangements are torsion-free. We also present a class of partitions for which Π_λ is not shellable, along with other issues scattered throughout the paper.     

Shellability of exponential structures

We show that the poset of all partitions of an nd-set with block size divisible by d is shellable. Using similar techniques, it also follows that various other examples of exponential structures cited by Stanley are also shellable. The method used involves the notion of recursive atom orderings introduced by Björner and Wachs.Research supported in part by NATO post-doctoral grant administered by the NSF.     

A new lattice construction

We introduce a new construction for orders and lattices. This construction is used to create large locally finite lattice varieties with uncountably many subvarieties.Dedicated to the memory of Ivan RivalReceived October 7, 2003; accepted in final form July 13, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Convex partitions of a finite lattice

The purpose of this paper is to introduce the lattice of convex partitions for a lattice L. Then we will show some properties of this lattice. Finally, we will show that if the convex partition lattice of L is finite and modular if and only if L is a finite chain. Presented by R. McKenzie. Received December 16, 2004; accepted in final form March 7, 2006.     

Ordered sets with no chains of ideals of a given type

We study the possible order types of chains of ideals in an ordered set. Our main result is this. Given an indecomposable countable order type , there is a finite listA ₁ , ...,A _n of ordered sets such that for every ordered setP the setJ(P) of ideals ofP, ordered by inclusion, contains a chain of type if and only ifP contains a subset isomorphic to one of theA ₁ ^#x03B1; , ...,A _n . The finiteness of the list relies on the notion of better quasi-ordering introduced by Nash-Williams and the properties of scattered chains obtained by Laver.The results presented. here constitute the second chapter of the third cycle thesis presented by the second author before the Claude Bernard University, Lyon (July 1983).     

Inequalities for the greedy dimensions of ordered sets

Every linear extension L: [x ₁<x ₂<...<x _m ] of an ordered set P on m points arises from the simple algorithm: For each i with 0i<m, choose x _i+1 as a minimal element of P–{x _j :ji}. A linear extension is said to be greedy, if we also require that x _i+1 covers x _i in P whenever possible. The greedy dimension of an ordered set is defined as the minimum number of greedy linear extensions of P whose intersection is P. In this paper, we develop several inequalities bounding the greedy dimension of P as a function of other parameters of P. We show that the greedy dimension of P does not exceed the width of P. If A is an antichain in P and |P–A|2, we show that the greedy dimension of P does not exceed |P–A|. As a consequence, the greedy dimension of P does not exceed |P|/2 when |P|4. If the width of P–A is n and n2, we show that the greedy dimension of P does not exceed n ²+n. If A is the set of minimal elements of P, then this inequality can be strengthened to 2n–1. If A is the set of maximal elements, then the inequality can be further strengthened to n+1. Examples are presented to show that each of these inequalities is best possible.Research supported in part by the National Science Foundation under ISP-80110451.Research supported in part by the National Science Foundation under ISP-80110451 and DMS-8401281.     

How small can a lattice of order-dimension n be?

The aim of this paper is to study the order-dimension of partition lattices and linear lattices. Our investigations were motivated by a question due to Bill Sands: For a lattice L, does dim L=n always imply |L|≥2 ⁿ ? We will answer this question in the negative since both classes of lattices mentioned above form counterexamples. In the case of the partition lattices, we will determine the dimension up to an absolute constant. For the linear lattice over GF(2), L _n , we determine the dimension up to a factor C/n for an absolute constant C.     

Some intersection theorems

LetL(A) be the set of submatrices of anm×n matrixA. ThenL(A) is a ranked poset with respect to the inclusion, and the poset rank of a submatrix is the sum of the number of rows and columns minus 1, the rank of the empty matrix is zero. We attack the question: What is the maximum number of submatrices such that any two of them have intersection of rank at leastt? We have a solution fort=1,2 using the followoing theorem of independent interest. Letm(n,i,j,k) = max(|F|;|G|), where maximum is taken for all possible pairs of families of subsets of ann-element set such thatF isi-intersecting,G isj-intersecting andF ansd,G are cross-k-intersecting. Then fori≤j≤k, m(n,i,j,k) is attained ifF is a maximali-intersecting family containing subsets of size at leastn/2, andG is a maximal2k?i-intersecting family. Furthermore, we discuss and Erd?s-Ko-Rado-type question forL(A), as well.     

Applications of the symmetric chain decomposition of the lattice of divisors

The symmetric chain decomposition of the lattice of divisors,D _N, is applied to prove results about the strict unimodality of the Whitney numbers ofD _N, about minimum interval covers for the union of consecutive rank-sets ofD _N, and about the distribution of sums of vectors in which each vector can be included several times (an extension of the famous Littlewood-Offord problem)Research supported in part by NSA/MSP GrantMDA904-92H3053.     

Weakening partial orderings to lattice and semilattice orderings

A partial ordering on a set P can be weakened to an upper or lower semilattice ordering, respectively a lattice ordering, if and only if P is filtered in the appropriate direction(s).This work was done while the author was partly supported by NSF contract DMS 85-02330.     

Greedy linear extensions for minimizing bumps

Let P be a finite poset and let L={x ₁<...n} be a linear extension of P. A bump in L is an ordered pair (x _i , x _i+1) where x _ii+1 in P. The bump number of P is the least integer b(P), such that there exists a linear extension of P with b(P) bumps. We call L optimal if the number of bumps of L is b(P). We call L greedy if x _{i j} for every j>i, whenever (x _i, x _i+1) is a bump. A poset P is called greedy if every greedy linear extension of P is optimal. Our main result is that in a greedy poset every optimal linear extension is greedy. As a consequence, we prove that every greedy poset of bump number k is the linear sum of k+1 greedy posets, each of bump number zero.This research (Math/1406/31) was supported by the Research Center, College of Science, King Saud University, Riyadh, Saudi Arabia.     

Irredundant self-dual bases for self-dual lattice varieties

For any finitely based self-dual variety of lattices, we determine the sizes of all equational bases that are both irredundant and self-dual. We make the same determination for {0, 1}-lattice varieties.Received July 11, 2002; accepted in final form August 27, 2004.     

Greedy posets for the bump-minimizing problem

A bump (x _i,x _i+1) occurs in a linear extension L={x ₁<...n} of a poset P, if x _ii+1 in P. L. is greedy if x _ij for every j>i, whenever (x _i x _i+1) in a bump in L. The purpose of this paper is to give a characterization of all greedy posets. These are the posets for which every greedy linear extension has a minimum number of bumps.This research (Math/1406/31) was supported by the Research Center, College of Science, King Saud University, Riyadh, Saudi Arabia.     

Towards an axiomatization of orderings

A set of six axioms for sets of relations is introduced. All well-known sets of specific orderings, such as linear and weak orderings, satisfy these axioms. These axioms impose criteria of closedness with respect to several operations, such as concatenation, substitution and restriction. For operational reasons and in order to link our results with the literature, it is shown that specific generalizations of the transitivity condition give rise to sets of relations which satisfy these axioms. Next we study minimal extensions of a given set of relations which satisfy the axioms. By this study we come to the fundamentals of orderings: They appear to be special arrangements of several types of disorder. Finally we notice that in this framework many new sets of relations have to be regarded as a set of orderings and that it is not evident how to minimize the number of these new sets of orderings.Symbol Table U universe (infinite countable) - D set of possible domains (finite and non-empty subsets of U) - R set of all considered relations - A empty relation on A - Id A identity relation on A - All A all relation on A - c complement operator (see Definition 2.1) - v converse operator (see Definition 2.1) - s symmetric part (see Definition 2.1) - asymmetric part (see Definition 2.1) - n non-diagonal part (see Definition 2.1) - r reflexive closure (see Definition 2.1) We gratefully acknowledge the support by the Co-operation Centre of Tilburg and Eindhoven Universities.     

Angle orders,regular n-gon orders and the crossing number

A finite poset P(X,<) on a set X={ x ₁,...,x _m} is an angle order (regular n-gon order) if the elements of P(X,<) can be mapped into a family of angular regions on the plane (a family of regular polygons with n sides and having parallel sides) such that x _ij if and only if the angular region (regular n-gon) for x _i is contained in the region (regular n-gon) for x _j. In this paper we prove that there are partial orders of dimension 6 with 64 elements which are not angle orders. The smallest partial order previously known not to be an angle order has 198 elements and has dimension 7. We also prove that partial orders of dimension 3 are representable using equilateral triangles with the same orientation. This results does not generalizes to higher dimensions. We will prove that there is a partial order of dimension 4 with 14 elements which is not a regular n-gon order regardless of the value of n. Finally, we prove that partial orders of dimension 3 are regular n-gon orders for n3.This research was supported by the Natural Sciences and Engineering Research Council of Canada, grant numbers A0977 and A2415.