Fuzzy Mathematics

In the mid-1960's I had the pleasure of attending a talk by Lotfi Zadeh at which he presented some of his basic (and at the time, recent) work on fuzzy sets. This led to my 1971 paper "Fuzzy groups", which became the starting point of an entire literature on fuzzy algebraic structures. The book deals with fuzzy graph theory, fuzzy topology, fuzzy geometry, and fuzzy abstract algebra. It presents the concepts of fuzzy mathematics with applications to engineering, computer science, and mathematics. In this second edition the chapter on geometry is expanded and contains results on the degree of adjacency of two regions and the degree of surroundedness of a region by another. Applications to digital polygons and image description are also given. Furthermore, the new edition includes results on image enhancement and thresholding by optimization of fuzzy compactness as well as results concerning a Hausdorff distance between fuzzy subsets. The most current work on the solution of nonlinear systems of fuzzy intersection equations of fuzzy singletons has been added to the chapter on algebra. The book is written with engineers and computer scientists in mind, but it can also serve as a research guide to interested mathematicians as it involves current results.