Oriented rank three matroids and projective planes
by Franz B. Kalhoff
Abstract:
Quite recently, Goodman, Pollack, Wenger and Zamfirescu have proven
two conjectures by Gr\"unbaum right, showing that any arrangement of
pseudolines in the plane can be embedded into a flat projective plane
and that there exists a universal topological projective plane in
which every arrangement of pseudolines is stretchable. By Folkman and
Lawrence's theorem, this plane contains every finite (simple) oriented
rank three matroid. In this paper, we will also consider embeddings
of oriented rank three matroids into topological projective planes,
but we will take a quite different viewpoint: We shall show, that
there exists a projective plane P that contains the combinatorial
geometry of every finite, orientable rank three matroid $M_n$, such
that any choice of orientations of the $M_n$ ($n$ ranging over the
natural numbers) extend to an orientation of $P$. Furthermore these
orientations correspond to archimedean orderings of $P$, hence the
reorientation classes of every finite rank three matroid can be
studied by the set of archimedian orderings of $P$. Since, by a
celebrated result of Priess-Crampe, any archimedian projective plane
can be completed and thus embedded into a flat projective plane,
our results yield another proof of Gr\"unbaum's conjectures and a new
proof of the rank three case of Folkman and Lawrence's theorem.
Key words and phrases:
Combinatorial geometries, oriented matroids, weak orientations,
matroids with coefficients, the Tutte group of a matroid, ordered
projective planes, ordered partial planes, order functions,
halforderings, ternary fields, the radical of a ternary field, linear
spaces, ordered linear spaces.
1991 Mathematics Subject Classification: Primary 05B35; Secondary 51G05, 15A15, 51A05, 51A10.
Available as
To appear in: European Journal of Combinatorics
Contact: Franz.Kalhoff@Mathematik.Uni-Dortmund.DE