On the use of ordered sets in problems of comparison and consensus of classifications

Ordered set theory provides efficient tools for the problems of comparison and consensus of classifications Here, an overview of results obtained by the ordinal approach is presented Latticial or semilatticial structures of the main sets of classification models are described Many results on partitions are adaptable to dendrograms; many results on n-trees hold in any median semilattice and thus have counterparts on ordered trees and Buneman (phylogenetic) trees For the comparison of classifications, the semimodularity of the ordinal structures involved yields computable least-move metrics based on weighted or unweighted elementary transformations In the unweighted case, these metrics have simple characteristic properties For the consensus of classifications, the constructive, axiomatic, and optimization approaches are considered Natural consensus rules (majoritary, oligarchic, ) have adequate ordinal formalizations A unified presentation of Arrow-like characterization results is given In the cases of n-trees, ordered trees and Buneman trees, the majority rule is a significant example where the three approaches convergeThe authors would like to thank the anonymous referees for helpful suggestions on the first draft of this paper, and W H E Day for his comments and his significant improvements of style