Lattice Diversities

TL;DR

This paper extends the concept of diversities from finite sets to arbitrary lattices, exploring their properties and limitations, especially regarding the tight span construction.

Contribution

It generalizes diversities to lattice structures and analyzes which properties are preserved and which are not in this broader context.

Findings

Many properties of diversities extend to lattice diversities.

The natural map to the tight span is not a lattice homomorphism.

A complete tight span theory does not develop as in metric and diversity cases.

Abstract

Diversities are a generalization of metric spaces, where instead of the non-negative function being defined on pairs of points, it is defined on arbitrary finite sets of points. Diversities have a well-developed theory. This includes the concept of a diversity tight span that extends the metric tight span in a natural way. Here we explore the generalization of diversities to lattices. Instead of defining diversities on finite subsets of a set we consider diversities defined on members of an arbitrary lattice (with a 0). We show that many of the basic properties of diversities continue to hold. However, the natural map from a lattice diversity to its tight span is not a lattice homomorphism, preventing the development of a complete tight span theory as in the metric and diversity cases.