Supersolvable and Modularly Complemented Matroid Extensions

Finite point configurations in projective spaces are combinatorially described by matroids, where full (finite) projective spaces correspond to connected modular matroids. Every representable matroid can in fact be extended to a modular one. However, we show that some matroids do not even have a modularly complemented extension. Enlarging the class of “ambient spaces” under consideration, we show that every matroid has a finite (but huge) supersolvable extension. In rank three, we prove that every matroid can be extended to a modularly complemented one — it is conjectured that one can even construct an extension that is modular (a finite projective plane).