A Universal Homogeneous Simple Matroid of Rank $3$

TL;DR

This paper constructs a universal, highly symmetric simple matroid of rank 3 that embeds all finite simple rank 3 matroids, explores its properties, and extends it to a projective plane with similar automorphism features.

Contribution

It introduces a universal homogeneous simple matroid of rank 3 and its variants, demonstrating their automorphism groups and properties, and extends these concepts to projective planes.

Findings

The constructed matroids are not -categorical.

They possess the independence property.

Their automorphism groups embed the symmetric group (64).

Abstract

We construct a $\wedge$-homogeneous universal simple matroid of rank $3$, i.e. a countable simple rank~$3$ matroid $M_*$ which $\wedge$-embeds every finite simple rank $3$ matroid, and such that every isomorphism between finite $\wedge$-subgeometries of $M_*$ extends to an automorphism of $M_*$. We also construct a $\wedge$-homogeneous matroid $M_*(P)$ which is universal for the class of finite simple rank $3$ matroids omitting a given finite projective plane $P$. We then prove that these structures are not $\aleph_0$-categorical, they have the independence property, they admit a stationary independence relation, and that their automorphism group embeds the symmetric group $Sym(\omega)$. Finally, we use the free projective extension $F(M_*)$ of $M_*$ to conclude the existence of a countable projective plane embedding all the finite simple matroids of rank $3$ and whose automorphism group contains $Sym(\omega)$, in fact we show that $Aut(F(M_*)) \cong Aut(M_*)$.