Kenneth P. Bogart

Preference structures I: Distances between transitive preference relations†

This paper generalizes the Kemeny‐Snell distance function for distances between weak orderings to a distance function for the collection of all partial orderings of a set. This generalization explains some of the seemingly strange properties of the Kemeny‐Snell distance, and extends it to such important classes of orderings as semiorders and interval orders. In section four I consider possible applications of the distance function and describe a number of problems that arise in attempts to apply the distance function. In section 3 I discuss the concept of a distance function in more general terms and introduce a new distance function defined by a set of axioms different from those given in Section 2.

A short proof that “proper = unit”

Abstract A short proof is given that the graphs with proper interval representations are the same as the graphs with unit interval representations.

Proper and unit tolerance graphs

We answer a question of Golumbic, Monma and Trotter by constructing proper tolerance graphs that are not unit tolerance graphs. An infinite family of graphs that are minimal in this respect is specified.

On the complexity of posets

The purpose of this paper is to discuss several invariants each of which provides a measure of the intuitive notion of complexity for a finite partially ordered set. For a poset X the invariants discussed include cardinality, width, length, breadth, dimension, weak dimension, interval dimension and semiorder dimension denoted respectively X, W(X), L(X), B(X), dim(X). Wdim(X), Idim(X) and Sdim(X). Among these invariants the following inequalities hold. B(X)=

A bound on the dimension of interval orders

We make use of the partially ordered set (I(0, n), <) consisting of all closed intervals of real numbers with integer endpoints (including the degenerate intervals with the same right- and left-hand endpoints), ordered by [a, b] < [c, d] if b < c, to show that there is no bound on the order dimension of interval orders. We then turn to the problem of computing the dimension of I(0, n), showing that I(0, 10) has dimension 3 but I(0, 11) has dimension 4. We use these results as initial conditions in obtaining an upper bound on the dimension of I(0, n) as a logarithmic function of n. It is our belief that this example is a “canonical” example for interval orders, so that the computation of its dimension should have significant impact on the problem of computing the dimension of interval orders in general.

An obvious proof of Fishburn's interval order theorem

Abstract This note gives a brief proof and slight generalization of Fishburns representation theorem for interval orders.

An elementary proof of the MacWilliams theorem on equivalence of codes

In this paper, we prove the following theorem due to MacWilliams. Two codes are equivalent if and only if there is a weight preserving linear isomorphism between them. The proof we give is a simplification and extension of her proof to the most general case.

Maximal dimensional partially ordered sets II. characterization of 2n-element posets with dimension n

In this paper, we show that if a partially ordered set has 2n elements and has dimension n, then it is isomorphic to the set of n-1 element subsets and 1element subsets of a set, ordered by inclusion, or else it has six elements and is isomorphic to a partially ordered set we call the chevron or to its dual.

Incidence codes of posets: Eulerian posets and Reed-Muller codes

This paper shows how to construct analogs of Reed-Muller codes from partially ordered sets. In the case that the partially ordered set is Eulerian the length of the code is the number of elements in the poset, the dimension is the size of a selected order ideal and the minimum distance is the minimum size of a principal dual ideal generated by a member of the order ideal. In this case, the majority logic method of decoding Reed-Muller codes works for incidence codes. A number of interesting combinatorial questions arise from the study of these codes.

Bipartite tolerance orders

Abstract Tolerance orders and tolerance graphs arise as a generalization of interval orders and interval graphs in which some overlap of intervals is tolerated. The classes of bounded, proper and unit tolerance orders and graphs have been studied by other authors. Here we introduce two-sided versions of these classes. We show that all the classes are identical for bipartite ordered sets and give some examples to show that the classes differ in the nonbipartite case. We also give an algorithm for determining whether a bipartite ordered set is in these classes, and if so finding a representation of it.