ABSTRACT
There are a variety of common discrete structures – graphs, directed graphs,
partial orders, lattices, hypergraphs, matroids – and some lesser known ones,
at least to combinatorialists – preorders, finite topologies, simplicial complexes
and multicomplexes. We are going to bring these together within a general
framework, and show that while each of the discrete structures has its own
merits, uses and advantages, it is the modeling of one structure by another, or
by a seemingly unrelated structure elsewhere within mathematics, that often
provides startlingly new insights. The alternate viewpoints suggest new modes
of attack on old problems and less traveled roads to explore.