Indistinguishability Operators, 2011
Modelling Fuzzy Equalities and Fuzzy Equivalence Relations
Studies in Fuzziness and Soft Computing Series, Vol. 260

Indistinguishability operators are essential tools in fuzzy logic since they fuzzify the concepts of equivalence relation and crisp equality. This book collects all the main aspects of these operators in a single volume for the first time. The stress is put on the study of their structure and the monograph starts presenting the different ways in which indistinguishability operators can be generated and represented. Special attention is paid to the Representation Theorem and the Sup-T product. Extensionality of fuzzy subsets is studied in detail and is related to their observability and to the granularity. The metric behaviour of indistinguishability operators and their connection with cluster analysis and hierarchical trees is established. Different ways to aggregate such operators are given as well as a number of methods to obtain transitive approximations of a fuzzy relation. Applications to approximate reasoning and to the study of fuzzy subgroups are also provided. The book ends with a chapter on finite-valued indistinguishability operators.

Introduction.- Granularity and Extensional Sets.- Isometries Between Indistinguishability Operators.- min-indistinguishability Operators and Hierarchical Trees.- Betweenness Relations.- Dimension and Basis.- Aggregation of Indistinguishability Operators.- Making Proximities Transitive.- Fuzzy Functions.- Indistinguishability Operators and Approximate Reasoning.- Vague Groups.- Finitely Valued Indistinguishability Operators.

Well organized book with very clear mathematical reasoning Helps to better understand the world of ideas related to imprecise partitionings and the major aspects of indistinguishability operators Written by a leading expert in the field