Matroids over hyperfields

TL;DR

This paper introduces a unified algebraic framework called matroids over hyperfields, generalizing various matroid concepts and establishing foundational properties and dualities within this new setting.

Contribution

It defines weak and strong matroids over hyperfields, provides cryptomorphic axiom systems, and proves duality theorems, advancing the theoretical understanding of matroid generalizations.

Findings

Weak and strong matroids over hyperfields are introduced.

Cryptomorphic axiom systems are established for these matroids.

Duality theorems are proved, and equivalence of weak and strong notions is shown for doubly distributive hyperfields.

Abstract

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Plucker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over F coincide.