Defining a (left) weak matroid over a skew hyperfield is easy enough, you just use your weak circuit axioms. The key to proving duality etc. is to replace the Grassmann-Plucker functions with Plucker ratio’s, as follows.

For any matroid $N$, let

$$A_N:=\{(B,B’): B, B’\text{ are adjacent bases of }N\}.$$

Consider a (skew) hyperfield $H$. Then $[.]:A_N\rightarrow H$ is a * left Plucker ratio map* if

**(P0)** $[Fa, Fb] [Fb, Fa]=1$

**(P1)** $[Fac,Fbc] [Fab, Fac] [Fbc, Fab]=1$

**(P2)** $[Fa,Fb] [Fb, Fc] [Fc, Fa]=-1$

**(P3)** $[Fac, Fbc]=[Fad, Fbd]$ whenever $Fab$ is not a basis.

**(P4)** $1\in [Fbd, Fab] [Fac, Fcd]+[Fad, Fab] [Fbc, Fcd]$

In the presence of an underlying matroid $N$, these axioms are cryptomorphic to the left circuit axioms (C0)-(C3). From any left $H$-matroid $M$ with circuits $\mathcal{C}$, one can derive a Plucker ratio map by setting

$$[Fa, Fb]_M:= X(a)^{-1}X(b)$$

for any circuit $X\in\mathcal{C}$ with $a,b\in \underline{X}\subseteq Fab$.

In the converse direction, a Plucker ratio map allows the reconstruction of $H$-oriented circuits $\mathcal{C}$ that satisfy the weak circuit axioms (given the underlying matroid $N$).

The dual of a left Plucker ratio map for $N$ is a right Plucker ratio map for $N^*$ determined by

$$[B, B’]^* = -[E-B, E-B’].$$

This allows to construct the dual $M^*$ of a weak left $H$-matroid $M$ via the Plucker ratio map.

The *cross ratio* is defined as $cr(F,a,b,c,d):=[Fac, Fbc][Fbd, Fad]$.

Only axiom (P4) refers to the hyperfield addition, and although I have not written it up for tracts I’m quite confident that the whole thing works in that setting as well. In the proofs I only ever refer to the multiplicative group of $H$ and the collection $$\{(a,b) \in H^2: 1\in a+b\}.$$

]]>I don’t think our result quite shows that, since we only forbid $5$- and $6$-tuples of nonbases in a specific structure; potentially we still keep rank-$4$ minors on eight elements with $> 4$ nonbases that are not Vamos-like.

However, I’ve thought about similar questions, and I think it should be the case that there is some constant $\alpha > 0$ so that there are a lot of matroids whose $m$-element minors all have at most $\alpha m$ nonbases; for $s \approx m/2$ this is very few indeed.

In fact, I think we can show, by sampling from the Graham/Sloane construction instead of the whole Johnson graph, that for a worse value of $\alpha$, the same statement holds when $n^{-2}$ is replaced with $n^{-h}$ for any $h$ strictly greater than $1$.

This is certainly eroding my belief in the conjecture that for sparse paving $N$, almost every matroid contains $N$ as a minor.

]]>The proof of your final claim shows that there are very many sp matroids (at least $c n^{-2}\binom{n}{n/2}$ in the exponent) so that each minor on $m=8$ elements of rank $s=4$ has at most $k=4$ nonbases. What function $k=k(m,s)$ makes this true in general?

(Point 2 just above that claim seems to have swallowed some surrounding text.)

]]>https://en.wikipedia.org/wiki/Dickson%27s_lemma

and

https://en.wikipedia.org/wiki/Buchberger%27s_algorithm

Dicksons lemma follows by a straightforward induction argument.

So no need to apply Zorns Lemma here.

]]>I think your interpretation of the ultraproduct is exactly right.

]]>But these other proofs (especially 2) are really elegant.

Proof #3 — which I don’t really understand — looks a lot like ‘the product partial field of infinitely many finite fields has a homomorphism to a field of char 0’.

]]>I think there may be an opportunity for some reverse mathematics here: which axiom scheme is necessary to prove Theorem 1?

]]>I was a bit confused by proof #2 at first, but I think you made a typo. Don’t you mean that if M has no representation in char. 0, then T + { *not* S_p : p prime } is inconsistent?

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