Matroids from Modules

Matroids from Modules
Nils Anders Danielsson and Michael B. Smyth
Electronic Notes in Theoretical Computer Science, Vol. 74, Elsevier 2003. Presented at MFCSIT2002. [pdf, ps.gz]

Abstract

The aim of this work is to show that (oriented) matroid methods can be applied to many discrete geometries, namely those based on modules over integral (ordered) domains. The trick is to emulate the structure of a vector space within the module, thereby allowing matroid methods to be used as if the module were a vector space. Only those submodules which are "closed under existing divisors," and hence behave like vector subspaces, are used as subspaces of the matroid.

It is also shown that Hübler's axiomatic discrete geometry can be characterised in terms of modules over the ring of integers.