John N. Mordeson Libri

This book presents an up-to-date account of research in important topics of fuzzy group theory. It concentrates on the theoretical aspects of fuzzy subgroups of a group. It includes applications to abstract recognition problems and to coding theory. The book begins with basic properties of fuzzy subgroups. Fuzzy subgroups of Hamiltonian, solvable, P-Hall, and nilpotent groups are discussed. Construction of free fuzzy subgroups is determined. Numerical invariants of fuzzy subgroups of Abelian groups are developed. The problem in group theory of obtaining conditions under which a group can be expressed as a direct product of its normal subgroups is considered. Methods for deriving fuzzy theorems from crisp ones are presented and the embedding of lattices of fuzzy subgroups into lattices of crisp groups is discussed as well as deriving membership functions from similarity relations. The material presented makes this book a good reference for graduate students and researchers working in fuzzy group theory.

Fuzzy graph theory is explored in depth, offering foundational knowledge for future applications across various fields. Key theories are introduced, including Craine's fuzzy interval graphs and Marczewski's theorem, alongside the Fulkerson and Gross characterization and Menger's theorem. The text emphasizes critical concepts like connectivity, distance, and saturation in fuzzy graphs, setting the stage for further exploration in an upcoming companion book by the same authors.

Lotfi Zadeh introduced the notion of a fuzzy subset of a set in 1965. Ris seminal paper has opened up new insights and applications in a wide range of scientific fields. Azriel Rosenfeld used the notion of a fuzzy subset to put forth cornerstone papers in several areas of mathematics, among other discplines. Rosenfeld is the father of fuzzy abstract algebra. Kuroki is re sponsible for much of fuzzy ideal theory of semigroups. Others who worked on fuzzy semigroup theory, such as Xie, are mentioned in the bibliogra phy. The purpose of this book is to present an up to date account of fuzzy subsemigroups and fuzzy ideals of a semigroup. We concentrate mainly on theoretical aspects, but we do include applications. The applications are in the areas of fuzzy coding theory, fuzzy finite state machines, and fuzzy languages. An extensive account of fuzzy automata and fuzzy languages is given in [100]. Consequently, we only consider results in these areas that have not appeared in [100] and that pertain to semigroups. In Chapter 1, we review some basic results on fuzzy subsets, semigroups, codes, finite state machines, and languages. The purpose of this chapter is to present basic results that are needed in the remainder of the book. In Chapter 2, we introduce certain fuzzy ideals of a semigroup, namely, fuzzy two-sided ideals, fuzzy bi-ideals, fuzzy interior ideals, fuzzy quasi ideals, and fuzzy generalized bi-ideals.

This ambitious exposition by Malik and Mordeson on the fuzzification of discrete structures serves as a solid foundational text on a key topic while providing a clear pathway to understanding basic fuzzy set concepts. The authors present their material in a traditional format, ensuring that all propositions and theorems are supported by rigorous proofs, yet the readability remains high. The inclusion of well-chosen examples effectively illustrates the often complex concepts discussed. Specific topics, such as fuzzy versions of decision trees, networks, graphs, and automata, are presented so clearly that even those not primarily focused on these areas may find themselves drawn to explore the material further. Encountering such a well-crafted book prompts reflections on one’s own writing, as well as thoughts on how this work, despite its distance from my own interests, could still connect with them. My interest in fuzzy set theory has long involved linking aspects of it with corresponding probability concepts, particularly through the idea of random sets, which serves as a key vehicle for this exploration.

The purpose of this book is to present a wide variety of types of fuzzy mathematics used in medical research and in the modelling of diagnostic systems. Some techniques from fuzzy mathematics include fuzzy relation equations, group decision making, abstract algebra, clustering methods, belief functions, fuzzy measures, evidence theory, Sugero integrals, abduction, automata theory, genetic algebras, and hypergraphs. Applications to medicine include medical diagnosis, bone mineral density, stroke pathogenesis, clinical monitoring, and neuronal cell-assemblies. The five basic algorithms for data analysis (clustering based on fuzzy equivalence relations, fuzzy c-means algorithm, solving a system of fuzzy relational equations, calculating fuzzy measures, and calculating the combined basic probabilistic measure) are explained in the appendix. Some of these algorithms are translated into the programming language C++.

Focusing on the intersection of Artificial Intelligence and the Sustainable Development Goals, this textbook addresses pressing global issues such as climate change, poverty, and biodiversity loss. It aims to educate undergraduate students on sustainability while maintaining accessible mathematics. The author provides exercises and an instructor's manual, preparing readers for deeper exploration of global challenges and related literature. Key topics include country progress on SDGs and the impact of AI on these critical issues, making it a comprehensive resource for understanding sustainability.

The narrative recounts the author's experiences and reflections on the development of fuzzy set theory, sparked by a talk from Lotfi Zadeh in the mid-1960s. The author shares a personal connection to the concept, having proposed similar ideas during graduate studies in the 1950s. Inspired by Zadeh's work on fuzzy convexity and later by C. L. Chang's paper, the author ventured into fuzzy algebra, culminating in the influential "Fuzzy groups" paper in 1971. This work laid the foundation for a growing field of fuzzy algebraic structures.

By applying the mathematics of uncertainty, this book evaluates how effectively countries are meeting the 17 Sustainable Development Goals (SDGs) established by the UN, particularly regarding climate change, human trafficking, and modern slavery. The authors introduce a novel mathematical framework to address the challenges posed by limited data, ranking countries on their progress and identifying gaps in the foster care system that contribute to human trafficking. It serves as a valuable reference for advanced students, researchers, and educators in mathematics and social sciences.

Advanced concepts in fuzzy graph theory are explored, particularly in their application to understanding and modeling illegal human trafficking. Building on foundational knowledge from a previous work, this book introduces innovative ideas like intuitionistic fuzzy graphs and concepts of independence and domination. It also covers directed fuzzy networks and incidence graphs, emphasizing practical applications of the theory. This comprehensive approach aims to enhance research in this evolving field.

Focusing on the application of fuzzy mathematics and graph theory, this book offers tools to address social issues like human trafficking and illegal immigration. It delves into advanced concepts such as connectivity, Wiener indices, and operations on fuzzy graphs using t-norms, while also exploring dialectic synthesis in this context. Each chapter presents key applications, making it an authoritative guide for both newcomers and experienced readers interested in the theory and modern applications of fuzzy graphs.