Separoids: A Mathematical Framework for Conditional Independence and Irrelevance

We introduce an axiomatic definition of a mathematical structure that we term a separoid. We develop some general mathematical properties of separoids and related axiom systems, as well as connections with other mathematical structures, such as distributive lattices, Hilbert spaces, and graphs. And we show, by means of a detailed account of a number of models of the separoid axioms, how the concept of separoid unifies a variety of notions of ‘irrelevance’ arising out of different formalisms for representing uncertainty in Probability, Statistics, Artificial Intelligence, and other fields.