Fuzzy Set Theory drew much of its initial inspiration and further developments to problems from pattern classification and cluster analysis. As it is pointed out in (Zadeh, 1977), this relationship rests, essentially, on the fact that in most real situations, the question is not whether a given object is or not a member of a class, but the degree in which the object belongs to the class; which amounts to saying that most classes in real situations are fuzzy in nature. In (Pal and Dutta, 1986) it is pointed out that such fuzzy nature of real world classification problems, may throw some light on the general problem of decision making in both random and non-random environment and explain by itself the great number of papers dealing with fuzzy techniques in speech and pattern recognition, image processing, cluster analysis and related topics.
In more concrete terms, there are some remarkable facts which support the above assertion, like the one-to-one correspondence which exists between dendograms (i.e. hierarchical classifications ) and similarity relations (i.e. fuzzy equivalence relations) and the existing duality between these relations and ultrametrics; which link up two of the basic entities involved in any clustering procedure, i.e. a similarity (or dissimilarity) among samples measured through a distance between them.
Ruspini, Bezdek and others made important developments in this context. Thus, on one hand, fuzzy partitions as well as classification algorithms yielding such partitions were introduced and, on the other hand, from the weakening of the ultrametric inequality arose a more general class of fuzzy binary relations, the so-called likeness relations which are in one-to-one correspondence with ordinary bounded metrics, making clearer still the duality between fuzzy transitive relations and metrics. More generally, as it was proven in (Valverde, 1985), such duality exists between reflexive, symmetric and T-transitive fuzzy relations, T being a continuous t-norm, and a kind of generalized pseudo-metrics over the unit interval; the so-called S-pseudometrics, S being a continuous t-conorm. This property is used to give a metrical characterization of fuzzy partitions, i.e. what can be recognized as the classifications associated with fuzzy equivalence relations. On the other hand, it is worth noticing that the family of classical classifications associated with general T-transitive relations - which, as we noticed, are the dendograms when T=Min - are the so-called overlapping stratified clusters (Jardine and Sibson, 1971), for which an additional structure is given by the T-transitive property or the associated S-pseudometric.
As it is well known, within a classical context, an equivalence relation in a set defines a partition or a classification in it, and viceversa. There have been several attempts to extend these concepts to the fuzzy framework, so that in the existing literature on this subject, two different trends have been followed. The first one puts its emphasis on the definition of fuzzy partition and then, studies the properties of the associated relation, if it exists. The papers by (Bezdek & Harris, 1978) and (Ovchinnikov & Riera, 1982) are representative of this research line. We follow an opposite path where the stress lies on the conditions that must fulfil a fuzzy partition in order to have an indistinguishability relation as its associated relation. In this trend we find the papers by (Ruspini, 1982), (Valverde, 1985) and (Jacas, Trillas & Valverde, 1987). In the first one by Ruspini, the concept of R-cluster and fuzzy R-cluster coverage is introduced as a convenient definition of classification associated to a likeness relation, such definition of R-cluster links up the relation R defined in the basic set X, with a "metric in the unit interval" that, in this case, is the restriction to this interval of the euclidean metric.
In (Valverde, 1985) these previous ideas have been collected and used
in order to obtain a metric characterization of fuzzy cluster coverages
associated to T-transitive relations. In the last section of this
work we present results which go one step further and characterize the
classifications associated to any given S-metric m in the
unit interval, showing that the m-cluster coverages defined in this
way are intrinsically linked to T-transitive relations. In fact,
the definition of m-cluster we introduce is a generalization of
the one given for classical clusters in terms of its characteristic functions,
where the underlying metric is the discrete distance in the two point set
.
The applicability of the above mentioned results drew, essentially, on the representation theorem (Valverde, 1985) for fuzzy T-transitive relations. As it is stressed in the next section, this theorem is a powerful tool to build fuzzy transitive relations starting from arbitrary fuzzy subsets of the given set. Such fuzzy subsets are called generators of the fuzzy relation and the third section is devoted to characterizing them. The fact that they appear as the closed sets associated with the fuzzy topology generated by the given relation may be considered as the most significative result concerned with the generators of a relation. An algorithm to compute a suitable family of such generators for similarity relations is given.
All the results described in this work are concerned with the following concepts that we recall here for sake of completeness; nevertheless, further details can be found, for instance, in (Trillas and Valverde, 1982), (Valverde, 1985) and (Jacas, 1988).
There is a close relation between T-indistinguishability relations and S- pseudometrics as is shown in the following theorem:
