Exterior algebras in matroid theory

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.

13 pages. v2: minor edits, addition of Remark 2.4. To appear in manuscripta math

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