Exterior algebras in matroid theory

TL;DR

This paper introduces an exterior algebra analogue for $unpm$-algebras within ordered blueprints, offering a new cryptomorphism for matroids and connecting to classical algebraic structures.

Contribution

It develops a novel exterior algebra framework for $unpm$-algebras, bridging matroid theory with algebraic structures like rings and semifields.

Findings

Provides a new cryptomorphism for matroids

Recovers classical exterior algebra from rings

Connects to Giansiracusa Grassmann algebra from semifields

Abstract

Ordered blueprints are algebraic objects that generalize monoids and ordered semirings, and $\mathbb{F}_1^{\pm}$-algebras are ordered blueprints that have an element $\epsilon$ that acts as $-1$. In this work we introduce an analogue of the exterior algebra for $\mathbb{F}_1^{\pm}$-algebras that provides a new cryptomorphism for matroids. We also show how to recover the usual exterior algebra if the $\mathbb{F}_1^{\pm}$-algebra comes from a ring, and the Giansiracusa Grassmann algebra if the $\mathbb{F}_1^{\pm}$-algebra comes from an idempotent semifield.