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.     

On circle containment orders

A partially ordered set is called acircle containment order provided one can assign to each element of the poset a circle in the plane so thatxy iff the circle assigned tox is contained in the circle assigned toy. It has been conjectured that every finite three-dimensional partially ordered set is a circle containment order. We show that the infinite three dimensional posetZ ³ isnot a circle containment order.Research supported in part by the Office of Naval Research, contract number N00014-85-K0622.Research supported in part by National Science Foundation, grant number DMS-8403646.     

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.     

Connectedness and diameter for random orders of fixed dimension

Let P _k (n) be the (partial) order determined by intersecting k random linear orderings of a set of size n; equivalently, let P _k (n) consist of n points chosen randomly and independently from the unit cube in ^k , with the induced product order. We show for each fixed k>1, that with probability approaching 1 as n, the comparability graph of P _k (n) is connected and has diameter 3.     

Interval orders and circle orders

A finite poset is an interval order if its point can be mapped into real intervals so that x in the poset precisely when x's interval lies wholly to the left of y's; the poset is a circle order if its points can be mapped into circular disks in the plane so that x precisely when x's circular disk is properly included in y's. This note proves that every finite interval order is a circle order.     

On the Boolean dimension of spherical orders

It is well known that if a planar order P is bounded, i.e. has only one minimum and one maximum, then the dimension of P (LD(P)) is at most 2, and if we remove the restriction that P has only one maximum then LD(P)3. However, the dimension of a bounded order drawn on the sphere can be arbitrarily large.The Boolean dimension BD(P) of a poset P is the minimum number of linear orders such that the order relation of P can be written as some Boolean combination of the linear orders. We show that the Boolean dimension of bounded spherical orders is never greater than 4, and is not greater than 5 in the case the poset has more than one maximal element, but only one minimum. These results are obtained by a characterization of spherical orders in terms of containment between circular arcs.Part of this work was carried out while both authors were visiting the Department of Applied Mathematics (KAM) of Charles University, Prague. The authors acknowledge support from the EU HCM project DONET.     

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.     

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.     

Degrees of freedom versus dimension for containment orders

Given a family of sets L, where the sets in L admit k degrees of freedom, we prove that not all (k+1)-dimensional posets are containment posets of sets in L. Our results depend on the following enumerative result of independent interest: Let P(n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that log P(n, k)nk log n where k is fixed and n is large.Research supported in part by Allon Fellowship and by a grant from Bat Sheva de Rothschild Foundation.Research supported in part by the Office of Naval Research, contract number N00014-85-K0622.     

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.     

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.     

Decomposition of partial orders

A decomposition theory for partial orders which arises from the split decomposition of submodular functions is introduced. As a consequence of this theory, any partial order has a unique decomposition consisting of indecomposable partial orders and certain highly decomposable partial orders. The highly decomposable partial orders are completely characterized. As a special case of partial orders, we consider lattices and distributive lattices. It occurs, that the highly decomposable distributive lattices are precisely the Boolean lattices.     

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.     

Edge subdivision and dimension

The dimension of a partial order can be multiplied by an arbitrarily large factor when edges are subdivided.This research was supported by National Science Foundation grant DCR-8604577.     

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.     

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.     

Some inequalities for partial orders

We establish some inequalities connecting natural parameters of a partial order P. For example, if every interval [a,b] contains at most maximal chains, if some antichain has cardinality v, and if there are ₁ chains whose union is cofinal and coinitial in P, then the chain decomposition number for P is ₁v (Theorem 2.2), and the inequality is sharp in a certain sense (Section 3).This paper was written while the authors were visitors at the Laboratoire d'algèbre ordinale, Département de Mathématiques, Université Claude Bernard, Lyon 1, France.Research supported by NSERC grant # A5198.     

Linear extensions of partial orders preserving monotonicity

Given a poset (A, r) and an acyclic r-monotone function f: AA, we prove that r can be extended to a linear order R with xRyf(x)Rf(y) for all x, yA.     

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.     

A note on the dimension of a poset

It is shown that the dimension of a poset is the smallest cardinal number such that there is an embedding of the poset into a strict product of linear orders.