Hopf algebras for matroids over hyperfields

TL;DR

This paper extends the theory of matroids over hyperfields by generalizing minors and direct sums, and constructs a Hopf algebra framework for these generalized matroids, unifying various matroid theories.

Contribution

It introduces generalized minors and direct sums for matroids over hyperfields and develops a corresponding Hopf algebra structure, broadening the scope of matroid theory.

Findings

Generalized minors and direct sums for hyperfield matroids

Construction of matroid-minor Hopf algebras over hyperfields

Unification of various matroid generalizations

Abstract

Recently, M.~Baker and N.~Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields.