Dwight Duffus

A structure theory for ordered sets

The theory of ordered sets lies at the confluence of several branches of mathematics including set theory, lattice theory, combinatorial theory, and even aspects of modern operations research. While ordered sets are often peripheral to the mainstream of any of these theories there arise, from time to time, problems which are order-theoretic in substance. The aim of this work is to fashion a classification scheme for ordered sets which aimed at providing a unified vantage point for some of the problems encountered with ordered sets. This classification scheme is based on a structure theory much akin to the familiar subdirect representation theory so useful in general algebra. The novelry of the structure theory lies in the importance that we attach to, and the widespread use that we make of, the concept of retract. At present, some vindication for our classification scheme can be found by examining its effectiveness for totally ordered sets (e.g., well-ordered sets, the rationals, reals, etc.) as well as for those finite ordered sets that arise commonly in combinatorial investigations (e.g., crowns and fences).

Fibres and ordered set coloring

Abstract A fibre F of a partially ordered set P is a subset which intersects each nontrivial maximal antichain of P. Let λ be the smallest constant such that each finite partially ordered set P contains a fibre of size at most λ |P|. We show that λ \ 2 3 by finding a good 3-coloring of the nontrivial antichains of P. Some decision problems involving fibres are also considered. For instance, it is shown that the problem of deciding if a partially ordered set has a fibre of size at most κ is NP-hard.

Exponentiation and Duality

In a unified treatment of cardinal and ordinal arithmetic G. Birkhoff [18], [20] defined (cardinal) exponentiation of ordered sets: for ordered sets X and Y, X Y (the power) is the set of all order-preserving maps of Y (the exponent) to X (the base) ordered componentwise. Our aim. is to review a significant body of results concerning powers of ordered sets and to present some central open problems arising in recent work. Roughly, we have two topics: first, an analysis of the structure of powers and their symmetries; second, a study of duality results for lattice-ordered algebras.

Two-colouring all two-element maximal antichains

Abstract A fibre in a partially ordered set P is a subset of P meeting every maximal antichain of P . We give an example of a finite poset P with no one-element maximal antichain and containing no fibre of size at most | P }2, thus answering a question of Aigner-Andreae and disproving a conjecture of Lonc-Rival. We also prove Theorem 1. The elements of an arbitrary partially ordered set can be coloured with two colours such that every two-element maximal antichain receives both colours .

An explicit 1-factorization in the middle of the Boolean lattice

Abstract An explicit definition of a 1-factorization of Bk (the bipartite graph defined by the k- and (k + 1)-element subsets of [2k + 1]), whose constituent matchings are defined using addition modulo k + 1, is introduced. We show that the matchings are invariant under rotation (mapping under σ = (1, 2, 3, …, 2k + 1)), describe the effect of reflection (mapping under p = (1, 2k + 1)(2, 2k)…(k, k + 2)), determine that there are no other symmetries which map these matchings among themselves, and prove that they are distinct from the lexical matchings in Bk.

Lexicographic matchings cannot form Hamiltonian cycles

For any positive integer k let B(k) denote the bipartite graph of k- and k+1-element subsets of a 2k+1-element set with adjacency given by containment. It has been conjectured that for all k, B(k) is Hamiltonian. Any Hamiltonian cycle would be the union of two (perfect) matchings. Here it is shown that for all k>1 no Hamiltonian cycle in B(k) is the union of two lexicographic matchings.

Dimension and automorphism groups of lattices

Every group is the automorphism group of a lattice of order dimension at most 4. We conjecture that the automorphism groups of finite modular lattices of bounded dimension do not represent every finite group. It is shown that ifp is a large prime dividing the order of the automorphism group of a finite modular latticeL then eitherL has high order dimension orMp, the lattice of height 2 and orderp+2, has a cover-preserving embedding inL. We mention a number of open problems.

Spanning retracts of a partially ordered set

Two general kinds of subsets of a partially ordered set P are always retracts of P: (1) every maximal chain of P is a retract; (2) in P, every isometric, spanning subset of length one with no crowns is a retract. It follows that in a partially ordered set P with the fixed point property, every maximal chain of P is complete and every isometric, spanning fence of P is finite.

Fixed points of products and the strong fixed point property

This problem motivates the present work: If ordered sets X and Y both have the fixed point property for order preserving maps has their product as well? Here we present a related condition — the so-called strong fixed point property — which arises from naive attempts to solve the problem. We are concerned with determining the nature and extent of this property. Several questions are raised concerning its relation to the fixed point property and other conditions such as dismantlability and contractibility.

Lattices arising in categorial investigation of Hedetniemi's conjecture

Abstract We discuss Hedetniemis conjecture in the context of categories of relational structures under homomorphisms. In this language Hedetniemis conjecture says that if there are no homomorphisms from the graphs G and H to the complete graph on n vertices then there is no homomorphism from G × H to the complete graph. If an object in some category has just this property then it is called multiplicative. The skeleton of a category of relational structures under homomorphisms forms a distributive lattice which has for each of the objects K of the category a pseudocomplementation. The image of the distributive lattice under such a pseudo-complementation is a Boolean lattice with the same meet as the distributive lattice and the structure K is multiplicative if and only if this Boolean lattice consists of at most two elements. We will exploit those general ideas to gain some understanding of the situation in the case of graphs and solve completely the Hedetniemi-type problem in the case of relational structures over a unary language.