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.     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

Minimizing setups in ordered sets of fixed width

A simply polynomial time algorithm is given for computing the setup number, or jump number, of an ordered set with fixed width. This arises as an interesting application of a polynomial time algorithm for solving a more general weighted problem in precedence constrained scheduling.     

A cofinal coloring theorem for partially ordered algebras

If P is a directed partially ordered algebra of an appropriate sort-e.g. an upper semilattice-and has no maximal element, then P has two disjoint subalgebras each cofinal in P. In fact, if P has cofinality then there exists a family of such disjoint subalgebras. A version of this result is also proved without the directedness assumption, in which the cofinality of P is replaced by an invariant which we call its global cofinality.This work was done while the first author was partly supported by NSF contract MCS 82-02632.     

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.     

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.     

Substitution and atomic extension on greedy posets

In this paper, we study the operations of substitution and atomic extension on greedy posets. For the substitution operation, if P=(P ₁ , x, P ₂ )is a greedy poset, then P ₁ and P ₂ are greedy posets, the converse being false. However, for the atomic extension, P=P ₁ (x, P ₂ )is a greedy poset if and only if P ₁ and P ₂ are greedy posets. We prove also that the class of greedy semi-partitive lattices is the smallest one containing M _n (n2), B ₃ and closed by atomic extension. The class C _n of greedy posets with jump number n is infinite. However, we show that C _n can be obtained, in a very simple way, from a subclass D _n of finite cardinal ity. We construct D _n for n=1, 2.     

Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

For a commutative ring R with set of zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we show that Γ(T(R)) and Γ(R) are isomorphic as graphs, where T(R) is the total quotient ring of R, and that Γ(R) is uniquely complemented if and only if either T(R) is von Neumann regular or Γ(R) is a star graph. We also investigate which cardinal numbers can arise as orders of equivalence classes (related to annihilator conditions) in a von Neumann regular ring.     

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,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.     

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.     

A sperner-type theorem

Let P=P ₁×P ₂×...×P _M be the direct product of symmetric chain orders P ₁, P ₂, ..., P _M . Let F be a subset of P containing no l+1 elements which are identical in M–1 components and linearly ordered in the Mth one. Then max |F|cM ^1/2lW(P), where W(P) is the cardinality of the largest level of P, and c is independent of P, M and l. Infinitely many P show that this result is best possible for every M and l apart from the constant factor c.     

A structural characterization of non-Arguesian lattices

This is the first of a planned series of papers on the structure of non-Arguesian modular lattices. Apart from the (subspace lattices of) non-Arguesian projective planes, the best known examples of such lattices are obtained via the Hall-Dilworth construction by badly gluing together two projective planes of the same order. Our principal result shows that every non-Arguesian modular lattice L retains some of the flavor of these examples: There exist in the ideal lattice of L 20 intervals, not necessarily distinct, that form non-degenerate projective plains, and 10 points and 10 lines in these planes that constitute in a natural sense a classical non-Arguesian configuration.Research supported by NSERC Operating Grant A8190.Research supported by NSF Grant DMS-8300107.     

A strange pigeon-hole principle

Using Ramsey theory, we establish the following pigeon-hole type principle: From a large number of random variables (functions, vectors, etc.) one can always select two, X and Y, such that P(X < Y) 1/2. We apply the principle for a poset problem.     

Identities of cofinal sublattices

If V is a variety of lattices and L a free lattice in V on uncountably many generators, then any cofinal sublattice of L generates all of V. On the other hand, any modular lattice without chains of order-type +1 has a cofinal distributive sublattice. More generally, if a modular lattice L has a distributive sublattice which is cofinal modulo intervals with ACC, this may be enlarged to a cofinal distributive sublattice. Examples are given showing that these existence results are sharp in several ways. Some similar results and questions on existence of cofinal sublattices with DCC are noted.This work was done while the first author was partly supported by NSF contract MCS 82-02632, and the second author by an NSF Graduate Fellowship.     

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.     

Tensorial decomposition of concept lattices

A tensor product for complete lattices is studied via concept lattices. A characterization as a universal solution and an ideal representation of the tensor products are given. In a large class of concept lattices which contains all finite ones, the subdirect decompositions of a tensor product can be determined by the subdirect decompositions of its factors. As a consequence, one obtains that the tensor product of completely subdirectly irreducible concept lattices of this class is again completely subdirectly irreducible. Finally, applications to conceptual measurement are discussed.Dedicated to Ernst-August Behrens on the occasion of his seventieth birthday.     

Random graph orders

Let P _n be the order determined by taking a random graph G on {1, 2,..., n}, directing the edges from the lesser vertex to the greater (as integers), and then taking the transitive closure of this relation. We call such and ordered set a random graph order. We show that there exist constants c, and °, such that the expected height and set up number of P _n are sharply concentrated around cn and °n respectively. We obtain the estimates: .565<c<.610, and .034<°<.289. We also discuss the width, dimension, and first-order properties of P _n.     

Scattered subsets of\mathbb{Q}

A linear order isscattered if it contains no copy of the rationals. The scattered subsets of form a better quasi-order under embeddability via order automorphisms.Research supported by NSERC grant #9848.