Diversity, Dependence and Independence

TL;DR

This paper introduces a unified framework for understanding dependence and independence using a rank-based approach, linking concepts across algebra, logic, and diversity theory.

Contribution

It presents a general theory of dependence and independence that encompasses various existing concepts through a rank-based formalism.

Findings

Dependence correlates with limited diversity

Independence corresponds to maximum diversity

The framework unifies algebraic and logical dependence concepts

Abstract

We introduce the concepts of dependence and independence in a very general framework. We use a concept of rank to study dependence and independence. By means of the rank we identify (total) dependence with inability to create more diversity, and (total) independence with the presence of maximum diversity. We show that our theory of dependence and independence covers a variety of dependence concepts, for example the seemingly unrelated concepts of linear dependence in algebra and dependence of variables in logic.