Friday, September 20, 2013

Incidence structures as height two orders

Incidence structures are defined by a set of elements and another set of edges over those elements such that each edge is dependent upon some subset of the set of elements. Every incidence structure can be expressed as an oriented set system with sets of edges grouped by their corresponding elements:
{#{1 2} #{4}, 
 #{0 1} #{3}}
Incidence structures may be represented as height two partial orders with an option to classify disconnected nodes as either empty nodes or empty edges. If we exclude empty edges as a possibility as we can also represent height two partial orders as incidence structures:

One advantage of such incidence structures is that they allow us to define mathematical substructures of a structured sets such as relations and hypergraphs as lower suborders of the height two partial order and they can likewise be used to describe all other parts of an incidence structure.

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