Exterior algebras in matroid theory

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.