Fuzzy semigroups.
When the concept of fuzzy set was introduced by \textit{L. A. Zadeh} in his seminal paper ``Fuzzy sets'' [Inf. Control 8, 338-353 (1965; Zbl 0139.24606)], it attracted the attention of some influential scientists and mathematicians. Theoreticians recognized that the concept has opened a new direction in pure mathematics. A. Rosenfeld is the first one to use the notion of a fuzzy subset in several areas of mathematics, especially in abstract algebras. After Kuroki's works on fuzzy ideal theory of semigroups, many authors have worked in the region of fuzzy semigroup theory. The purpose of this book is to present an up to date account of fuzzy subsemigroups and fuzzy ideals of a semigroup, it is an aspect of algebraic semigroup theory. Some applications in fuzzy coding theory, fuzzy finite state machines, and fuzzy languages are also included in this book. The book consists of 10 Chapters. After some basic results on fuzzy subsets, semigroups, codes, finite state machines and languages, in Chapter 2, certain fuzzy ideals of a semigroup are discussed. In Chapter 3, Some fuzzy regular semigroups, for example, left (right) regular semigroups, complete regular semigroups, quasi-regular semigroups, are characterized in terms of fuzzy ideals of a semigroup. In Chapter 4, the semilattices of groups are considered mainly in terms of fuzzy ideals. In Chapter 5, the relationship of fuzzy congruences on a group and fuzzy normal subgroups is established. In Chapters 6, 7 \(T^*\)-pure semigroups, fuzzy congruences on a \(T^*\)-pure semigroup and different types of fuzzy prime ideals are considered. Applications are given in Chapters 8, 9, 10 including fuzzy coding on a free monoid, fuzzy finite machines and regular fuzzy languages which did not appear in another book written by \textit{J. N. Mordeson} and \textit{D. S. Malik} [Fuzzy automata and languages. Theory and applications (2002; Zbl 1046.68068)]. The book represents a major contribution to the literature on fuzzy semigroups and its applications in fuzzy coding, fuzzy finite state machines and fuzzy languages. They are indispensable for researchers in this field, but also highly suitable as textbooks for students at the graduate level.
scientific article; zbMATH DE number 1980895