Matroids over partial hyperstructures

TL;DR

This paper introduces a unified algebraic framework for various matroid theories using new objects called tracts, generalizing existing structures like hyperfields and partial fields, and explores their properties and dualities.

Contribution

The paper defines matroids over tracts, introduces cryptomorphic axiom systems, and establishes conditions under which weak and strong matroids coincide, generalizing multiple matroid concepts.

Findings

Matroids over tracts unify various matroid theories.

Weak and strong matroids coincide over doubly distributive partial hyperfields.

Examples show differences between weak and strong matroids in this framework.

Abstract

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which generalize both hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. We then define matroids over tracts; in fact, there are (at least) two natural notions of matroid in this general context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Pl\"ucker functions, and dual pairs, and establish some basic duality results. We then explore sufficient criteria for the notions of weak and strong matroids to coincide. For example, if $F$ is a particularly nice kind of tract called a doubly distributive partial hyperfield, we show that the notions of weak and strong $F$-matroids coincide. We also give examples of tracts $F$ and weak $F$-matroids which are not strong. Our theory of matroids over tracts is closely related to, but more general than, "matroids over fuzzy rings" in the sense of Dress and Dress-Wenzel.