A New Tool to Analyze Partially Ordered Sets Application: ranking of polychlorinated biphenyls and alkanes/alkenes in river Main, Germany

Partial order renders a helpful tool in many ranking problems. Especially when multiindicator systems come into play, the product order as a special order relation is an adequate mathematical structure. A typical outcome of partial order analysis are the chains, where the values of the indicators are weak monotonously increasing. However, one also finds in partial order analysis incomparable elements. In this paper we suggest the construction of a special type of a binary relation, a tripartite graph. By means of the tripartite graph the role of indicators causing incomparabilities can be clarified. Furthermore, quantities derived from tripartite graphs help to decide whether or not the sets of incomparable elements are topologically connected. Hence, tripartite graphs enlighten the interpretation of complex data sets.