The residuation model for the ordinal construction of dissimilarities and other valued objects

The aim of this chapter is to present an ordinal model of valued objects, special cases of which appear in many contexts. The model lies on basic notions of ordered set theory: residuation or, equivalently, Galois connections; it is not new: explicitly proposed in fuzzy set theory by Achache (1982, 1988), it also underlies an order formalization of a Jardine and Sibson (1971) model given by Janowitz (1978; see also Barthélemy, Leclerc and Monjardet 1984a). Here, our main concern is to apply the model in order to obtain and study dissimilarities such as ultrametrics, Robinson or tree-compatible ones. Valued objects of other types, already considered in the literature, will be also given as examples: two types of valued non symmetric relations and two types of valued convex subsets. The chapter is neither a theoretical general presentation nor a detailed study of a few special cases. It is, tentatively, something between these extreme points of view. Some references are given to the reader interested to more details on a specific class of valued objects, or to more information about residuation (or Galois mappings) theory. In what follows, E will be a given finite set with n elements. Several families of combinatorial objects defined on E will be considered.