Finite projective geometry and the Krasner hyperfield

Question

The Krasner hyperfield is an algebraic structure of two operations on $K=\{0,1\}$ called $+\colon K\times K\to \mathcal{P}(K)$ and $\cdot\colon K\times K\to K$ with

• $0+0=0$
• $0+1=1+0=1$
• $1+1=\{0,1\}$

and with the same multiplicative structure as $\mathbb{F}_2$. In Proposition 3.1 of this paper of Connes and Consani, it is shown that projective geometries in which every line contains at least 4 points are "essentially the same" as $K$-modules. What I am trying to understand is the way in which this connects to ordinary vector spaces. For instance, if I look at $\mathbb{F}_p^n$ then, as a vector space, it determines a projective geometry (via, if you like, a matroid) which is a module over $K$. But $\mathbb{F}_p^n$ is also just a module over $\mathbb{F}_p$. I believe that $K$ is the terminal hyperfield, so there's a hyperfield morphism $\mathbb{F}_p\to K$. If I take a projective geometry, realized as a module over $K$, what is the pullback along the terminal morphism? Does the resulting $\mathbb{F}_p$ vector space have the same "projective geometry," abstractly, as the combinatorial one we started with?

Answer 1

A module over $\mathbb{F}_p$ viewed as a hyperfield is different than that of a module of $\mathbb{F}_p$ viewed as a ring. In particular the pullback of a module over the Krasner hyperfield need not be a vector space over $\mathbb{F}_p$. In fact it is never an $\mathbb{F}_p$ vector space since since it must obey the rule $x+x= \{0,x\}$ for all elements $x \neq 0$ in the module over $K$.