The Graph of Bases of the 2-Dimensional Integral Lattice
Stefan Schmidt
MIT
To any ring R there is attached a (possibly infinite)
graph G(R) in a natural way: The edges of G(R) are the bases of
(the left R-module) RxR and the set-theoretic union of all edges
defines the vertice set of G(R). In case R is a field then clearly
G(R) is connected and has diameter at most 2. Remarkably, the same
holds whenever R is any finite ring (like the ring of integers
modulo n).
In our talk we investigate G(Z) where Z is the ring of integers.
We first observe that G(Z) is connected and then ask: "What is
the diameter of G(Z)?" The answer is a little surprising---but
quite elementary---and therefore will be given in full detail!
CONE May 2000