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.