The Diagram

Abstract

Pictures nurture and prompt mathematical discovery. Yet in mathematical exposition these pictures are often gone. The rough notes and the pictures are forsaken for the polish. Happily the modern theory of ordered sets has no such inhibitions. Pictures and pictorial aids abound in the literature. There are schematic illustrations intended to highlight one aspect without faithfully replicating others, time diagrams commonly used in the theory and practice of scheduling, block diagrams in orthomodular lattices. By far the most common scheme is a graphical one, and the one best understood is the comparability graph, which assigns a vertex to each element of the ordered set and an edge joins a pair of vertices whenever they are related by comparability. Nevertheless, the graphical scheme in most common use is the diagram. And while this artifice has been long in use it is little understood. Lately this theme is attracting more attention. Besides the challenge of the unsolved one reason for the vitality of the diagram theme lies in its potential for highlighting graphical configurations of use both in combinatorial and structural problems.