Fuzzy mathematics. An introduction for engineers and scientists
This is an interesting and nicely written book which covers on a mathematically rather elementary level (no category-theoretic methods are used, and no formal logic tools) standard topics of applicationally relevant parts of algebra and geometry, often illustrating the notions with simple examples. Chapter 1, ``Fuzzy subsets'', mainly covers fuzzy relations, their basic properties, and some applications, for example, to pattern classification. Chapter 2 treats fuzzy graphs, understood as fuzzy (symmetric) relations in fuzzy sets (over finite universes of discourse). It discusses notions such as connectedness, clusters, and operations with (sub-)graphs, and considers applications to cluster analysis, fuzzy intersection equations, and database theory. The following three chapters treat geometrical matters. This starts in Chapter 3, ``Fuzzy topological spaces'', which introduces standard notions of (usual) topological and metric spaces, gives their fuzzification essentially à la \textit{C. L. Chang} [J. Math. Anal. Appl. 24, 182-190 (1968; Zbl 0167.51001)], and discusses, for example, continuity of mappings, compactness of fuzzy spaces, and iterations of mappings of fuzzy topological spaces into themselves. These tools are the background for the discussion of fuzzy digital topology in Chapter 4. Again, the main notions of classical digital topology are first explained, and then generalized to the fuzzy case with emphasis on matters such as connectedness, surroundedness, components, and convexity. The third member of this group, Chapter 5 on fuzzy geometry, treats fuzzified elementary plane geometry as developed in a series of papers by A. Rosenfeld. Topics include the area and the perimeter of a fuzzy subset of the Euclidean plane, distances between fuzzy sets, fuzzy rectangles, triangles, points and lines, and fuzzy projective geometry. The final Chapter 6 is devoted to fuzzy abstract algebra, and discusses topics such as substructures, monoids, groups, and rings in their fuzzified versions. (Again a section on the corresponding crisp notions starts the chapter.) Applications concern topics such as coding theory, formal power series and automata, as well as equations in polynomial rings. Reviewer's remarks: The book is quite readable, although the reviewer regrets the practice of giving references locally at the end of each chapter instead of globally at the end of the book.
scientific article; zbMATH DE number 1259213
scientific article