Ranking regions using cluster analysis, Hasse diagram technique and topology

We developed the theoretical background of the application of Hierarchical Cluster Analysis HCA to the improvement of the understanding of Hasse Diagrams coming from the ranking process developed for the application of the Hasse Diagram Technique HDT. The use of HCA is based on the idea of the reduction of the number of elements considered in a poset (partially ordered set) by the selection of representatives from the original set to rank. We showed that the clusters arisen from HCA can be interpreted as similarity classes where one of its members (the nearest to the centre of the cluster) can be selected as representative of its class. We applied this procedure to the reduction of a set of 59 regions of Baden Wuerttemberg, Germany, monitored with respect to Pb, Cd, Zn and S pollution in the herb layer. After applying HCA we found 26 representatives and we drew the Hasse diagram of this set of representatives. Finally, we describe the mathematical procedure to endow a partially ordered set with a topology taking advantage of the network structure of a Hasse diagram where the comparisons of the elements are considered as the open sets of a topological basis. Using such a basis we calculated the closure of two subsets of the set of 26 regions (big traffic or sensitive ecosystems and industrialised regions). We showed how can be interpreted the results of the closures of these subsets