Projective Geometries and Simple Pointed Matroids as $\mathbb{F}_1$-modules

TL;DR

This paper constructs a fully faithful embedding of projective geometries and simple pointed matroids into $un$-modules, generalizing classical embeddings and connecting geometric and algebraic structures.

Contribution

It introduces a new embedding of projective geometries into $un$-modules via hypermagmas, extending classical algebraic and geometric frameworks.

Findings

Embedding of projective geometries into $un$-modules is fully faithful.

Generalizes Eilenberg-MacLane and Segal's nerve constructions.

Provides a new perspective on the algebraic structure of geometries.

Abstract

We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into $\mathbb{F}_1$-modules, in the sense of Connes and Consani. This factors through a faithful functor out of simple pointed matroids. This follows from our construction of a fully faithful embedding of weakly unital, commutative hypermagmas into $\fun$-modules. This embedding is of independent interest as it generalizes the classical Eilenberg-MacLane embedding for commutative monoids and recovers Segal's nerve construction for commutative partial monoids. For this reason, we spend some time elaborating its structure.