Circle orders,N-gon orders and the crossing number

Let ={P ₁,...,P _m } be a family of sets. A partial order P(, <) on is naturally defined by the condition P _i <P _j iff P _i is contained in P _j . When the elements of are disks (i.e. circles together with their interiors), P(, <) is called a circle order; if the elements of are n-polygons, P(, <) is called an n-gon order. In this paper we study circle orders and n-gon orders. The crossing number of a partial order introduced in [5] is studied here. We show that for every n, there are partial orders with crossing number n. We prove next that the crossing number of circle orders is at most 2 and that the crossing number of n-gon orders is at most 2n. We then produce for every n4 partial orders of dimension n which are not circle orders. Also for every n>3, we prove that there are partial orders of dimension 2n+2 which are not n-gon orders. Finally, we prove that every partial order of dimension 2n is an n-gon order.This research was supported under Natural Sciences and Engineering Research Council of Canada (NSERC Canada) grant numbers A2507 and A0977.     

Angle orders

A finite poset is an angle order if its points can be mapped into angular regions in the plane so thatx precedesy in the poset precisely when the region forx is properly included in the region fory. We show that all posets of dimension four or less are angle orders, all interval orders are angle orders, and that some angle orders must have an angular region less than 180° (or more than 180°). The latter result is used to prove that there are posets that are not angle orders.The smallest verified poset that is not an angle order has 198 points. We suspect that the minimum is around 30 points. Other open problems are noted, including whether there are dimension-5 posets that are not angle orders.Research supported in part by the National Science Foundation, grant number DMS-8401281.     

Angle orders and zeros

An angle order is a partially ordered set whose points can be mapped into unbounded angular regions in the plane such that x is less than y in the partial order if and only if x's angular region is properly included in y's. The zero augmentation of a partially ordered set adds one point to the set that is less than all original points. We prove that there are finite angle orders whose augmentations are not angle orders. The proof makes extensive use of Ramsey theory.     

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.     

Circle orders and angle orders

A poset is a circle order if its points can be mapped into circular disks in the plane so that x in the poset precisely when x's circular disk is properly included in y's; the poset is an angle order if its points can be mapped into unbounded angular regions that preserve < by proper inclusion. It is well known that many finite angle orders are not circle orders, but has been open as to whether every finite circle order is an angle order. This paper proves that there are finite circle orders that are not angle orders.     

Sphere orders

Ann-sphere order is a finite partially ordered set representable by containment ofn-spheres in Euclidean (n+1)-space. We present a sequence {P _i } of ordered sets such that eachP _i is ann-sphere order only forni; one consequence is that we are able to determine the dimension of a Euclidean space-time manifold from the finite suborders of its causality order.Research supported by ONR grant N00014 85-K-0769.     

Extremal problems for linear orders on [n] p

Let be a product order on [n] ^p i.e. for A, B [n] ^p , 1a ₁<a ₂<...<a _p º-n and 1<-b ₁<b ₂<...<b _p <-n we have AB iff a _i<-b _i for all i=1, 2,..., p. For a linear extension < of (ordering [n] ^p as ) let F _< ^[n],p (m) count the number of A _i 's, i<-m such that 1A _i. Clearly, for every m and <, where <l denotes the lexicographic order on [n] ^p . In this note we prove that the colexicographical order, <c, provides a corresponding lower bound i.e. that holds for any linear extension < of .This project together with [2] was initiated by the first author and continued in colaboration with the second author. After the death of the first author the work was continued and finalized by the second and the third author.Research supported by NSF grant DMS 9011850.     

Fractional dimension of partial orders

Given a partially ordered setP=(X, ), a collection of linear extensions {L ₁,L ₂,...,L _r } is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL _i (andy<x in someL _j ). For a positive integerk, we call a multiset {L ₁,L ₂,...,L _t } ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL _i 's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask. We prove various results about the fractional dimension of a poset.Research supported in part by the Office of Naval Research.     

A family of (n−1)! Diagonal polynomial orders ofN n

Letn>0 be an element of the setN of nonnegative integers, and lets(x)=x ₁+...+x _n , forx=(x ₁, ...,x _n ) N _n . Adiagonal polynomial order inN _n is a bijective polynomialp:N _n N (with real coefficients) such that, for allx,y N _n ,p(x)<p(y) whenevers(x)<s(y). Two diagonal polynomial orders areequivalent if a relabeling of variables makes them identical. For eachn, Skolem (1937) found a diagonal polynomial order. Later, Morales and Lew (1992) generalized this polynomial order, obtaining a family of 2 ^n–2 (n>1) inequivalent diagonal polynomial orders. Here we present, for eachn>0, a family of (n – 1)! diagonal polynomial orders, up to equivalence, which contains the Morales and Lew diagonal orders.     

Random orders

Letk andn be positive integers and fix a setS of cardinalityn; letP _k (n) be the (partial) order onS given by the intersection ofk randomly and independently chosen linear orders onS. We begin study of the basic parameters ofP _k (n) (e.g., height, width, number of extremal elements) for fixedk and largen. Our object is to illustrate some techniques for dealing with these random orders and to lay the groundwork for future research, hoping that they will be found to have useful properties not obtainable by known constructions.Supported by NSF grant MCS 84-02054.     

Cycle-free partial orders and chordal comparability graphs

This paper studies a number of problems on cycle-free partial orders and chordal comparability graphs. The dimension of a cycle-free partial order is shown to be at most 4. A linear time algorithm is presented for determining whether a chordal directed graph is transitive, which yields an O(n ²) algorithm for recognizing chordal comparability graphs. An algorithm is presented for determining whether the transitive closure of a digraph is a cycle-free partial order in O(n+m _t)time, where m _tis the number of edges in the transitive closure.     

Matchings and radon transforms in lattices. I. Consistent lattices

An element in a lattice is join-irreducible if x=ab implies x=a or x=b. A meet-irreducible is a join-irreducible in the order dual. A lattice is consistent if for every element x and every join-irreducible j, the element xj is a join-irreducible in the upper interval [x, î]. We prove that in a finite consistent lattice, the incidence matrix of meet-irreducibles versus join-irreducibles has rank the number of join-irreducibles. Since modular lattices and their order duals are consistent, this settles a conjecture of Rival on matchings in modular lattices.     

The many faces of circle orders

A finite partially ordered set P is called a circle order if one can assign to each x P a circular disk C _x so that xy iff C _x C _y . It is interesting to observe that many other classes of posets, such as space-time orders, parabola orders, the Loewner order for 2×2 Hermitian matrices, etc. turn out to be exactly circle orders (or their higher dimensional analogues). We give a global proof for these equivalences.Research supported in part by the Office of Naval Research, the Air Force Office of Scientific Research and by DIMACS.     

An Erdös-Ko-Rado theorem for the subcubes of a cube

LetP be that partially ordered set whose elements are vectors x=(x ₁, ...,x _n ) withx _i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx _i =y _i orx _i =0 for alli. LetN _i (P)={x εP : |{j:x _j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN ₁(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f ₀, ...,f _n ) for which there is an Erdös-Ko-Rado familyF inP such that |N _i (P) ∩F|=f _i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉.     

Spherical containment and the Minkowski dimension of partial orders

The recent work on circle orders generalizes to higher dimensional spheres. As spherical containment is equivalent to causal precedence in Minkowski space, we define the Minkowski dimension of a poset to be the dimension of the minimal Minkowski space into which the poset can be embedded; this isd if the poset can be represented by containment with spheresS ^d–2 and of no lower dimension. Comparing this dimension with the standard dimension of partial orders we prove that they are identical in dimension two but not in higher dimensions, while their irreducible configurations are the same in dimensions two and three. Moreover, we show that there are irreducible configurations for arbitrarily large Minkowski dimension, thus providing a lower bound for the Minkowski dimension of partial orders.     

On the automorphism conjecture for products of ordered sets

I. Rival and A. Rutkowski conjectured that the ratio of the number of automorphisms of an arbitrary poset to the number of order-preserving maps tends to zero as the size of the poset tends to infinity. We prove this hypothesis for direct products of arbitrary posets P=S ₁××S _n under the condition that max_i|S_i|=0(n/logn).     

Random orders of dimension 2

A relationship is established between (partially) ordered sets of dimension 2 chosen randomly on a labelled set, chosen randomly by isomorphism type, or generated by pairs of random linear orderings. As a consequence we are able to determine the limiting probability (in each of the above sample spaces) that a two-dimensional order is rigid, is uniquely realizable, or has uniquely orientable comparability graph; all these probabilities lie strictly between 0 and 1. Finally, we show that the number of 2-dimensional (partial) orderings of a labelled n-element set is .On leave from Emory University, Atlanta, GA. Research at Emory supported by ONR grant N00014 85-K-0769.     

Regressions and monotone chains: A ramsey-type extremal problem for partial orders

Aregression is a functiong from a partially ordered set to itself such thatg(x)≦x for allz. Amonotone k-chain is a chain ofk elementsx ₁<x ₂ <...<x _k such thatg(x ₁)≦g(x ₂)≦...≦g(x _k ). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it will have a monotone (k + 1)-chain. Fixingw, letf(w, k) be the smallest number such that every regression on every partial order with size leastf(w, k) but no antichain larger thanw has a monotone (k + 1)-chain. We show thatf(w, k)=(w+1) ^k . Dedicated to Paul Erdős on his seventieth birthday Research supported in part by the National Science Foundation under ISP-80-11451.     

New results from an algorithm for counting posets

In this paper we report on the success of a new technique for computing the number of unlabeled partial orders on n elements based on the partial order of partial orders ordered by containment. In addition to the number of partial orders, we obtain complete enumerations of the number of partial orders on n elements with r relations for n<-11, where r takes on all possible values. We point out some interesting sequences that arise in these tables.Supported by Natural Sciences and Engineering Research Council Grant No. OGP8053.