Automated Lattice Drawing

Abstract

Abstract. Lattice diagrams, known as Hasse diagrams, have played an ever increasing role in lattice theory and fields that use lattices as a tool. Initially regarded with suspicion, they now play an important role in both pure lattice theory and in data representation. Now that lattices can be created by software, it is important to have software that can automatically draw them. This paper covers:-The role and history of the diagram.-What constitutes a good diagram.-Algorithms to produce good diagrams. Recent work on software incorporating these algorithms into a drawing program will also be covered.An ordered set P = (P, ≤) consists of a set P and a partial order relation ≤ on P . That is, the relation ≤ is reflexive (x ≤ x), transitive (x ≤ y and y ≤ z imply x ≤ z) and antisymmetric (x ≤ y and y ≤ x imply x = y). If P is finite there is a unique smallest relation ≺, known as the cover or neighbor relation, whose transitive, reflexive closure is ≤. (Graph theorists call this the transitive reduct of ≤.) A Hasse diagram of P is a diagram of the acyclic graph (P, ≺) where the edges are straight line segments and, if a < b in P, then the vertical coordinate for a is less than the one for b. Because of this second condition arrows are omitted from the edges in the diagram.A lattice is an ordered set in which every pair of elements a and b has a least upper bound, a ∨ b, and a greatest lower bound, a ∧ b, and so also has a Hasse diagram.These Hasse diagrams 1 are an important tool for researchers in lattice theory and ordered set theory and are now used to visualize data. This paper deals the special issues involved in such diagrams. It gives several approaches that have been used to automatically draw such diagrams concentrating on a three dimension force algorithm especially adapted for ordered sets that does particularly well.We begin with some examples.1 In the second edition of his famous book on lattice theory [3] Birkhoff says these diagrams are called Hasse diagrams because of Hasse's effective use of them but that they go back at least to H. Vogt, Résolution algébrique deséquation, Paris, 1895.

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Hasse Diagrams of Combined Traces

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“…However, we wish to briefly present two tools for visualizing small lattices. Ralph Freese, University of Hawaii [4], has created a tool for visualizing small lattices, in which he positions the elements of the lattice by using an analogue positioning method. The method uses tractive and repulsive forces until the lattice has reached a stable condition.…”