A Similarity-Based Generalization of Fuzzy Orderings

Many fuzzy systems in real-world applications make implicit use of two fundamental concepts - similarity and ordering. For both of them, formulations in the framework of fuzzy relations have been proposed already in the early days of fuzzy set theory. While similarity relations have turned out to be very useful tools for the interpretation of fuzzy partitions and fuzzy controllers, the utilization of fuzzy orderings in more applied areas is still lying far behind. The main objective of this thesis is to find reasons for this missing link and, consequently, to present and investigate an alternative approach to fuzzy orderings which overcomes the problems in terms of applicability. First of all, the need for expressing orderings in vague environments is motivated with several examples. After providing the necessary preliminaries from the theory of fuzzy sets and relations, we turn to a critical view on the existing approaches to fuzzy orderings. By the help of three case studies, the definition of fuzzy antisymmetry turns out to be the crucial point. Resting upon this discovery, a generalization of fuzzy orderings is presented which also takes the strong relationship between similarity and ordering into account. The key idea is to replace the crisp equality in the definition of reflexivity and antisymmetry by a similarity relation. The remaining thesis is devoted to three topics. Firstly, constructions, representations, and characterizations of the new class of fuzzy orderings and their dual relations are studied in detail. Secondly, we investigate properties and representations of hulls with respect to fuzzy orderings which can be particularly useful for applications. Finally, a general framework for ordering fuzzy sets is introduced which can be applied to any domain for which a crisp or fuzzy ordering is known.