Supersolvable and modularly complemented matroid extensions

3. Thomas Wanner, Günter M. Ziegler:
   Supersolvable and modularly complemented matroid extensions
   European Journal of Combinatorics 12(4), pp. 341-360, 1991.

Abstract

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 3, 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).

Links

The published version of the paper can be found at https://doi.org/10.1016/S0195-6698(13)80117-5.

Bibtex

@article{wanner:ziegler:91a,
   author = {Thomas Wanner and G\"unter M. Ziegler},
   title = {Supersolvable and modularly complemented matroid extensions},
   journal = {European Journal of Combinatorics},
   year = 1991,
   volume = 12,
   number = 4,
   pages = {341--360},
   doi = {10.1016/S0195-6698(13)80117-5}
   }