and to put them into a wider context. Each chapter concludes with historical and bibliographical remarks that help the reader to orientate himself in the rapidly growing literature on fuzzy topics as well as to understand better the motivations of the field. Even a reader without the knowledge of modern algebra may gain by reading these remarks as well as the introductory parts of each chapter.
Chapter one discusses the basic concepts of fuzzy sets and mappings in an algebraic setting. The notions of "L-fuzzy set" and "Flou-set" are compared and their equivalence proved. The chapter continues with the discussion of probability defined on fuzzy events and with fuzzification of probabilities. It concludes with a brief exposition of categories of fuzzy sets.
Chapter two, entitled "Fuzzy theories" deals with various ways of the fuzzification of mathematical structures. Starting with fuzzification of categories it goes to fuzzification of topological spaces and concludes by a comparison of various ways in which mathematical structures can be fuzzified. Although brief and sketchy this chapter will make the reader aware of the 1'act that mathematical structures can be fuzzified in many distinct ways that are not necessarily equivalent.
Although Chapter three is entitled "Fuzzy logic", this part deals primarily with the description of some normal forms and with the minimization of many-valued (fuzzy) switching circuits. It is not an attempt to present a logic of vague reasoning. This chapter is rather disappointing for everyone who is familiar with many-valued logics. It has to be regretted that the authors did not take the opportunity to present at least some of the results of G. Moisirs school on algebraic logic and switching algebras that are otherwise available only in Roumanian or French.
Chapter four on fuzzy systems presents a brief algebraic treatment of reachability, and observability that has been motivated mainly by articles ofArbib & Zeigler and Arbib & Manes. There is not much on stability theory and the authors themselves state that their definition of stability is not fully adequate. The chapter also contains some original results of the authors concerning "deformed sets" and fuzzification of linear systems. The chapter ends with fuzzy systems in a category.
Chapter five deals with fuzzy automata, languages and algorithms. Readers will find it useful to have the most important definitions and concepts summarized on relatively few pages. The authors introduce the concept of semiring and a matrix over a lattice. This approach helps to unify the topics discussed here and to compare them with the material on crisp systems.
Chapter six (Deciding in fuzzy environment) presents the extension of some elementary concepts of mathematical programming to fuzzy sets (e.g. a convex fuzzy set etc.). However, the reader looking for some applicable results will be rather disappointed.
Chapter seven on fuzzy clustering has been surpassed by the rapid development of the theory as well as of practical applications since 1974 when the Roumanian original of this book was published.