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Fuzzy systems --- Fuzzy sets --- Systèmes flous --- Ensembles flous --- Fuzzy sets. --- Fuzzy systems. --- Systems, Fuzzy --- System analysis --- Fuzzy logic --- Sets, Fuzzy --- Fuzzy mathematics --- Set theory

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Decision making --- Marketing --- Investments --- Fuzzy sets --- Prise de décision --- Investissements --- Ensembles flous --- Fuzzy sets. --- Mathematical models --- Modèles mathématiques --- Mathematical models. --- Sets, Fuzzy --- Fuzzy mathematics --- Set theory --- Investing --- Investment management --- Portfolio --- Finance --- Disinvestment --- Loans --- Saving and investment --- Speculation --- Consumer goods --- Domestic marketing --- Retail marketing --- Retail trade --- Industrial management --- Aftermarkets --- Selling --- Deciding --- Decision (Psychology) --- Decision analysis --- Decision processes --- Making decisions --- Management --- Management decisions --- Choice (Psychology) --- Problem solving

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This book presents an up-to-date account of research in important topics of fuzzy group theory. The book concentrates on the theoretical aspects of fuzzy subgroups of a group. It also includes applications to some abstract recognition problems and to coding theory. The book begins with basic properties of fuzzy subgroups. The notions of ascending series and descending series of fuzzy subgroups are used to define nilpotency of a fuzzy subgroup. 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. The number of fuzzy subgroups (up to an equivalence relation) of certain finite Abelian groups is determined. 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. Deriving membership functions from similarity relations is considered. The material presented in this book makes it a good reference for graduate students and researchers working in fuzzy group theory.

Fuzzy sets. --- Group theory. --- Ensembles flous --- Groupes, Théorie des --- Engineering. --- Artificial intelligence. --- Systems and Information Theory in Engineering. --- Artificial Intelligence (incl. Robotics). --- Group Theory and Generalizations. --- Fuzzy sets --- Computer Science --- Algebra --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Théorie des groupes --- Sets, Fuzzy --- Groups, Theory of --- Substitutions (Mathematics) --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Construction --- Computer science. --- Computers. --- Complexity, Computational. --- Computer Science. --- Theory of Computation. --- Complexity. --- Information theory. --- Artificial Intelligence. --- Industrial arts --- Technology --- Communication theory --- Communication --- Cybernetics --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Computational complexity. --- Complexity, Computational --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Computer systems --- Calculators --- Cyberspace

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In 1965 Lotfi Zadeh, a professor of electrical engineering at the University of California in Berkeley, published the first of his papers on his new Fuzzy Set Theory. Since the 1980s this mathematical theory of "unsharp amounts" has been applied in many different fields with great success. The word "fuzzy" has also become very well-known among non-scientists thanks to extensive advertising campaigns for fuzzy-controlled household appliances and to their prominent presence in the media, first in Japan and then in other countries. On the other hand, the story of how Fuzzy Set Theory and its earliest applications originated remains largely unknown. In this book, the history of Fuzzy Set Theory and the ways it was first used are incorporated into the history of 20th century science and technology. Influences from philosophy, system theory and cybernetics stemming from the earliest part of the 20th century are considered alongside those of communication and control theory from mid-century. Today, Fuzzy Set Theory is the core discipline of "soft computing," and provides new impetus for research in the field of Artificial Intelligence.

Fuzzy sets. --- Set theory. --- Ensembles flous --- Théorie des ensembles --- Fuzzy sets --- Set theory --- Algebra --- Applied Mathematics --- Civil Engineering --- Engineering & Applied Sciences --- Mathematics --- Civil & Environmental Engineering --- Physical Sciences & Mathematics --- Aggregates --- Classes (Mathematics) --- Ensembles (Mathematics) --- Mathematical sets --- Sets (Mathematics) --- Theory of sets --- Sets, Fuzzy --- Computer science. --- Artificial intelligence. --- Computer-aided engineering. --- Applied mathematics. --- Engineering mathematics. --- Computer Science. --- Computer-Aided Engineering (CAD, CAE) and Design. --- Appl.Mathematics/Computational Methods of Engineering. --- Artificial Intelligence (incl. Robotics). --- Logic, Symbolic and mathematical --- Fuzzy mathematics --- Computer aided design. --- Mathematical and Computational Engineering. --- Artificial Intelligence. --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Engineering --- Engineering analysis --- Mathematical analysis --- CAD (Computer-aided design) --- Computer-assisted design --- Computer-aided engineering --- Design --- CAE --- Data processing

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This collection of articles is devoted to fuzzy as well as rough set theories. Both theoriesarebasedonrigorousideas,methodsandtechniquesinlogic,mathem- ics, and computer science for treating problems for which approximate solutions are possible only, due to their inherent ambiguity, vagueness, incompleteness, etc. Vast areas of decision making, data mining, knowledge discovery in data, approximatereasoning,etc., aresuccessfully exploredusing methods workedout within fuzzy and rough paradigms. By the very nature of fuzzy and rough paradigms, outlined above, they are related to distinct logical schemes: it is well-known that rough sets are related to modal logicsS5andS4(Orl owska, E., Modal logics in the theory of infor- tion systems, Z. Math. Logik Grund. Math. 30, 1984, pp. 213 ?.; Vakarelov, D., Modal logics for knowledgerepresentationsystems,LNCS 363,1989,pp. 257?.) and to ?nitely-valued logics (Pagliani, P., Rough set theory and logic-algebraic structures. In Incomplete Information: Rough Set Analysis,Orlo wska, E., ed., Physica/Springer, 1998, pp. 109 ?.; Polkowski, L. A note on 3-valued rough logic accepting decision rules, Fundamenta Informaticae 61, to appear). Fuzzy sets are related to in?nitely-valued logics (fuzzy membership to degree r? [0,1]expressingtruthdegreer)(Goguen,J.A.,Thelogicofinexactconcepts, Synthese18/19,1968-9,pp.325?.;Pavelka,J.,OnfuzzylogicI,II,III,Z. Math. Logik Grund. Math. 25, 1979, pp. 45 ?., pp. 119 ?., pp. 454 ?.; Dubois, D., Prade, H., Possibility Theory, Plenum Press, 1988; Haj ´ ek, P., Metamathematics of Fuzzy Logic, Kluwer, 1998).

Rough sets --- Fuzzy sets --- Ensembles approximatifs --- Ensembles flous --- Fuzzy sets. --- Rough sets. --- Artificial intelligence. --- Data mining. --- Information theory. --- Computer science. --- Database management. --- Artificial Intelligence. --- Data Mining and Knowledge Discovery. --- Theory of Computation. --- Computation by Abstract Devices. --- Mathematical Logic and Formal Languages. --- Database Management. --- Informatics --- Science --- Communication theory --- Communication --- Cybernetics --- Data base management --- Data services (Database management) --- Database management services --- DBMS (Computer science) --- Generalized data management systems --- Services, Database management --- Systems, Database management --- Systems, Generalized database management --- Electronic data processing --- Algorithmic knowledge discovery --- Factual data analysis --- KDD (Information retrieval) --- Knowledge discovery in data --- Knowledge discovery in databases --- Mining, Data --- Database searching --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Computers. --- Mathematical logic. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Computer systems --- Calculators --- Cyberspace --- Machine theory. --- Formal Languages and Automata Theory. --- Abstract automata --- Abstract machines --- Automata --- Mathematical machine theory --- Algorithms --- Logic, Symbolic and mathematical --- Recursive functions --- Robotics