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On the Evolution of the Mathematics of Fuzziness

  • Chapter
Fuzzy Set Theory and Advanced Mathematical Applications

Part of the book series: International Series in Intelligent Technologies ((ISIT,volume 4))

Abstract

Since the publication in 1965 of the seminal paper “Fuzzy Sets” by L.A. Zadeh, mathematicians started to fuzzify the classical mathematical structures. Three stages can be considered: firstly the straightforward fuzzification during the seventies, secondly the explosion of the possible choices in the generalization process during the eighties and currently the standardization, axiomatization and L-fuzzification. In this contribution we will illustrate these three stages by many examples and we will outline the stimulating role of fuzzy set theory in the expansion of mathematics. Most of the illustrative examples in the different stages in the development of the mathematics of fuzziness are taken from the results of our research group during the past twenty years.

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References

  1. L. Zadeh, “Fuzzy Sets”, Information Control 8 (1965), 338–353.

    Article  MathSciNet  MATH  Google Scholar 

  2. E.E. Kerre, “Basic principles of fuzzy set theory for the representation and manipulation of imprecision and uncertainty”, in: Introduction to the Basic Principles of Fuzzy Set Theory and some of its Applications (E.E. Kerre, ed.) Communication and Cognition, Gent, 1993, second revised edition, 1–158.

    Google Scholar 

  3. A. Rosenfeld, “Fuzzy groups”, J. Math. Anal. Appl. 35 (1971), 512–517.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Negoita, D. Ralescu, “Application of Fuzzy sets to System Analysis”,Birkhäuser Verlag, Basel, 1975.

    Google Scholar 

  5. C.L. Chang, “Fuzzy topological spaces”, J. Math. Anal. Appl. 24 (1988), 182–190.

    Article  Google Scholar 

  6. R. Lowen, “Fuzzy topological spaces and fuzzy compactness”, J. Math. Anal. Appl. 56 (1976), 621–633.

    Article  MathSciNet  MATH  Google Scholar 

  7. E.E. Kerre, “Fuzzy topologizing with preassigned operations”, International Congress for Mathematicians, Helsinki, 1978, 61–62.

    Google Scholar 

  8. P.M. Pu, Y.M. Liu, “Fuzzy topology I”, J. Math. Anal. Appl. 76 (1980), 571–599.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Sugeno, Theory of fuzzy integrals and its applications, PhD Thesis, Tokyo Institute of Technology, Tokyo, 1974.

    Google Scholar 

  10. B. Schweizer and A. Sklar, “Associative functions and statistical triangle inequalities”,Publ.Math. Debrecen 8 (1961), 169–186.

    MathSciNet  MATH  Google Scholar 

  11. R. Bellman, L. Zadeh, M. Giertz, “On the analytic formalism of the theory of fuzzy sets”, Information Sciences 5 (1973), 149–156.

    Article  MATH  Google Scholar 

  12. D. Dubois, H. Prade, “New results about properties and semantics of fuzzy-set-theoretic operators”, in: Fuzzy Sets, Theory and Application to Policy Analysis and Information Systems (P.P. Wang, S.K. Chang, eds.) Plenum Press, New York, 1980, 59–75.

    Google Scholar 

  13. R. Yager, “On the implication operator in fuzzy logic”, Information Sciences 31 (1983), 141–164.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Zimmermann, P. Zysno, “Latent connectives in human decision making”, Fuzzy Sets and Systems 4 (1980), 37–51.

    Article  MATH  Google Scholar 

  15. W. Bandler, L. Kohout, “Fuzzy power sets and fuzzy implication operators”, Fuzzy Sets and Systems 4 (1980), 13–30.

    Article  MathSciNet  MATH  Google Scholar 

  16. B. Hutton, “Normality in fuzzy topological spaces”, J. Math. Anal. Appl. 50 (1975), 74–79.

    Article  MathSciNet  MATH  Google Scholar 

  17. E.E. Kerre, “Characterization of normality in fuzzy topological spaces”, Simon Stevin, 53 (1979), 239–248.

    MathSciNet  MATH  Google Scholar 

  18. E.E. Kerre, P. Ottoy, “Fuzzy subspace of a fuzzy topological space”, in: Proceedings NAFIPS’ 88 (S. Ovchinnikov, ed.) San Francisco, 1988, 141–146.

    Google Scholar 

  19. E.E. Kerre, “Fuzzy Sierpinski space and its generalizations”, J. Math. Anal. Appl. 74 (1980), 318–324.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Goguen, “L-fuzzy sets”, J. Math. Anal. Appl. 18 (1967), 145–174.

    Article  MathSciNet  MATH  Google Scholar 

  21. C.K. Wong, “Fuzzy points and local properties of fuzzy topology”, J. Math. Anal. Appl. 46 (1974), 316–318.

    Article  MathSciNet  MATH  Google Scholar 

  22. E.E. Kerre, P. Ottoy, “On the different notions of neighbourhood in Chang-Goguen fuzzy topological spaces”, Simon Stevin 61 (1987), 131–146.

    MathSciNet  MATH  Google Scholar 

  23. H. Ludescher, E. Roventa, “Sur les topologies floues définies à l’aide des voisinages”, C.R. Acad. Sc. Paris t. 283 (1976) Serie A, 575–577.

    MathSciNet  MATH  Google Scholar 

  24. R.H. Warren, “Fuzzy topologies characterized by neighbourhood systems”, Rocky Mt. J. Math. 9 (1979), 761–764.

    Article  MathSciNet  MATH  Google Scholar 

  25. M.E. Abd. El-Monsef, M.H. Ghanim, E.E. Kerre, A.S. Mashhour, “Fuzzy topological results”, in: Proceedings Fifth Prague Topological Symposium (J. Novak, ed.) Helderman Verla, Berlin, 1983,1–5.

    Google Scholar 

  26. M. Ghanim, E.E. Kerre, A. Mashhour, “Separation axioms, subspaces and sums in fuzzy topology”, J. Math. Anal. Appl. 102 (1984), 189–202.

    Article  MathSciNet  MATH  Google Scholar 

  27. E.E. Kerre, P. Ottoy, “On the characterization of a Chang fuzzy topology by means of a Kerre neighbourhood system”, in: Proceedings NAFIPS 87 (J.L. Chameau, J. Yao, eds.), Purdue University Pres, Purdue, 1987, 302–307.

    Google Scholar 

  28. E.E. Kerre, P. Ottoy, “Lattice properties of neighbourhood systems in Chang fuzzy topological spaces”, Fuzzy Sets and Systems 30 (1989), 205–213.

    Article  MathSciNet  MATH  Google Scholar 

  29. E.E. Kerre, P. Ottoy, “A comparison of the different notions of neighbourhood systems for Chang topologies”, in: Proceedings of the First Joint IFSA-EC EURO-WG Workshop on Progress in Fuzzy Sets in Europe, (J. Kacprzyck, A. Straszak, eds.), Praceibspan, Warsaw, vol. 169, 1989, 241–251.

    Google Scholar 

  30. E.E. Kerre, P. Ottoy, “On α-generated fuzzy topologies”, Fasciculi Mathematici 19 (1990), 127–134.

    MathSciNet  MATH  Google Scholar 

  31. E.E. Kerre, P. Ottoy, “Counterexamples in fuzzy topology”, to appear in: Proceedings of the Third Polish Symposium on Fuzzy Sets and Interval Analysis in Pure and Applied Mathematics (J. Albrycht, ed.) Poznan, 1989.

    Google Scholar 

  32. M. Mizumoto, K. Tanaka, “Some properties of fuzzy numbers”, in: Advances in Fuzzy Set Theory and Applications (Gupta, Ragade, Yager, eds.) North Hollan, Amsterdam, 1979,153–164.

    Google Scholar 

  33. D. Dubois, H. Prade, Fuzzy Sets and Systems, Theory and Applications, Academic Press, New York, 1980.

    MATH  Google Scholar 

  34. S. Rodabaugh, “Fuzzy addition in the L-fuzzy real line”, Fuzzy Sets and Systems 8 (1982), 39–52.

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Bortolan, R. Degani, “A review of some methods for ranking fuzzy subsets”, Fuzzy Sets and Systems 15 (1985), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  36. E.E. Kerre, “The use of fuzzy set theory in electrocardiological diagnostics”, in: Approximate Reasoning in Decision Analysis (M. Gupta, E. Sanchez, eds.) North-Hollan, Amsterdam, 1982, 277–282.

    Google Scholar 

  37. E.E. Kerre, “Fuzzy-set-theoretic approach to medical diagnosis problems”, in: Proceedings Second World Conference on Mathematics at the Service of Man (A. Ballester, D. Cardus, E. Trillas, eds.) Universidad Politechnica de Las Palmas, Las Palmas, 1982, 388–393.

    Google Scholar 

  38. E.E. Kerre, “Fuzzy approach to EGG diagnosis”, in: Encyclopedia of Systems and Control (M. Singh, ed.) Pergamon Pres, Oxford, 1987,1405–1407.

    Google Scholar 

  39. E.E. Kerre, A. Van Schooten, “A deeper look on fuzzy numbers from a theoretical as well as a practical point of view”, in: Fuzzy Logic in Knowledge-Based Systems, Decision and Control (M. Gupta, T. Yamakawa, eds.) North-Hollan, Amsterdam, 1988,173–196.

    Google Scholar 

  40. R. Baekeland, E.E. Kerre, “Remarkable continuity-preserving properties of the extended operations on fuzzy numbers”, in: Proceedings 2nd joint IFSA-EC EURO-WG Workshop on Progress in Fuzzy Sets in Europe (H. Hansen, W. Janko, eds.) Wirtschäftuniversität, Wien, 1988, 28–31.

    Google Scholar 

  41. R. Baekeland, E.E. Kerre, “Piecewise linear fuzzy quantities: a way to implement fuzzy information into expert systems and fuzzy databases”, in: Uncertainty and Intelligent Systems, Lecture Notes in Computer Science no. 313, Springer-Verla, Berlin, 1988, 119–126.

    Google Scholar 

  42. H. Steyaert, F. Van Parys, R. Baekeland, E.E. Kerre, “A C-implementation of the algebraic operations on piecewise linear fuzzy quantities”, in: Advances in Fuzzy Sets and Applications (H. Teodorescu, I. Tofan, J.G. Aluya, 0. Costinescu, eds.) Editura Universitat, Iasi, 1992, 87–93.

    Google Scholar 

  43. H. Steyaert, F. Van Parys, R. Baekeland, E.E. Kerre, “The implementation of piecewise linear fuzzy quantities”, submitted to International Journal of Intelligent Systems, paper n° 9249 (1992).

    Google Scholar 

  44. R. Baekeland, E.E. Kerre, “Operations on piecewise linear fuzzy quantities: a theoretical approach”, submitted to Fuzzy Sets and Systems (1992).

    Google Scholar 

  45. B. De Baets, E.E. Kerre, “Analytical inference results for triangular fuzzy data”, in: Proceedings of the Second International Conference on Fuzzy Logic and Neural Networks, (T. Yamakawa, ed.), FLSI, Tokyo, 1992, 295–300.

    Google Scholar 

  46. B. De Baets, E.E. Kerre, M. Gupta, “Expert knowledge representation by means of piecewise linear fuzzy quantities”, in: Proceedings Third IFSA Congres, Seattle, 1989, 618–621.

    Google Scholar 

  47. X. Wang, E.E. Kerre, B. Cappelle, D. Ruan, “Transitivity of fuzzy orderings based on pairwise comparisons”, in: Fuzzy Logic and Intelligent Technologies in Nuclear Science (D. Ruan, A.O., eds.) World Scientific Publishing Compan, Singapore, 1994, 38–43.

    Google Scholar 

  48. W. Bandler, L. Kohout, “Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and artificial systems”, in: Fuzzy Sets Theory and Application to Policy Analysis and Information Systems (P.P. Wang, S.K. Chang, eds.) Plenum Press, New York, 1980, 341–367.

    Google Scholar 

  49. R. Zenner, E.E. Kerre, R. De Caluwe, “Practical determination of document description relations in document retrieval systems”, in: Proceedings Workshop on the Membership function, (C. Carlsson, ed.) EIASM, Brussels, 1984, 127–138.

    Google Scholar 

  50. R. Zenner, E.E. Kerre, R. De Caluwe, “A new approach to information retrieval systems using fuzzy expressions”, Fuzzy Sets and Systems 17 (1985), 9–22.

    Article  MathSciNet  MATH  Google Scholar 

  51. E.E. Kerre, R. Zenner, R. De Caluwe, “The use of fuzzy set theory in information retrieval and databases: a survey”, JASIS 37 (1986), 341–345.

    Google Scholar 

  52. R. Vandenberge, E.E. Kerre, A. Van Schooten, R. De Caluwe, “A practical application of fuzzy database techniques to criminal investigation”, in: Proceedings Second IFSA Congress, Tokyo, 1987, 661–664.

    Google Scholar 

  53. E.E. Kerre, P. Ottoy, “On the fuzzification of multivalued mappings”, in: Proceedings of the Workshop on Knowledge-Based Systems and Models of Logical Reasoning, Scomi-Scientific Center, F.A. El-Cheri, Cairo (1988).

    Google Scholar 

  54. R. Vandenberge, A. Van Schooten, R. De Caluwe, E.E. Kerre, “Some practical aspects of fuzzy database techniques: an example”, Information Sciences 14, no. 6 (1989), 465–472.

    Google Scholar 

  55. E.E. Kerre, “Fuzzy relational calculus in terms of after-and foresets and its application to information retrieval”, in: Proceedings of Assuit First International Conference on Mathematics and Statistics, Part VIII (Al-Hussaini, Abou El-Ela, eds.) Assuit, 1990, pp. 133–157.

    Google Scholar 

  56. E.E. Kerre, “A walk through fuzzy relations and their application to information retrieval, medical diagnosis and expert systems”, in: Proceedings of ISUMA ’90 (Bilal M. Ayyub, ed.), IEEE Computer Society Press, Los Alamito, CA, 1990, 84–89.

    Google Scholar 

  57. G. Chen, E.E. Kerre, J. Vandenbulcke, “A step towards the theory of fuzzy relational database design”, in: Proceedings IFSA 91, (B. Lowen, M. Roubens, eds.) Brussels, 1991, 44–47.

    Google Scholar 

  58. E.E. Kerre, “A walk through fuzzy relations and their applications”, in: Analysis and Management of Uncertainty (B. Ayyub, M. Gupta, L. Kanal, eds.) North-Hollan, Amsterdam, 1992,141–151.

    Google Scholar 

  59. B. De Baets, E.E. Kerre, “Some reflections on fuzzy relational compositions”, in: Proceedings IPMU 92 (B. Bouchon, R. Yager, eds.) Palma de Mallorca, 1992, 251–254.

    Google Scholar 

  60. B. De Baets, E.E. Kerre, “Triangular fuzzy relational compositions revisited”, in: Uncertainty in Intelligent Systems (B. Bouchon, L. Valverde, R. Yager, eds.) North-Hollan, Amsterdam, 1993, 257–268.

    Google Scholar 

  61. B. De Baets, E.E. Kerre, “A revision of Bandler-Kohout compositions of relations”, Mathematica Pannonica 4/1 (1993), 1–21.

    Google Scholar 

  62. B. De Baets, E.E. Kerre, “FRESH, A Pascal library for solving fuzzy relational equations on the unit interval”, in: Proceedings EUFIT 93, Aachen, 1993, 537–541.

    Google Scholar 

  63. G. Chen, E.E. Kerre, J. Vandenbulcke, “A general treatment of data redundancy in a fuzzy relational data model”, JASIS 43, no. 4 (1992), 304–311.

    Article  Google Scholar 

  64. G. Chen, E.E. Kerre, J. Vandenbulcke, “Fuzzy functional dependency and axiomatic system in a fuzzy relational data model”, in: Proceedings IPMU 92 (B. Bouchon, R. Yager, eds.) Palma de Mallorca, 1992, 313–316.

    Google Scholar 

  65. G. Chen, E.E. Kerre, J. Vandenbulcke, “The axiomatic system of fuzzy functional dependency in a fuzzy relational data model”, in: Information and Systems 92 (Zhang Shengkai, Zou Kaiqi, eds.) Dalian Maritime University Publishing Hous, Dalian, 1992, 876–880.

    Google Scholar 

  66. G. Chen, E.E. Kerre, J. Vandenbulcke, “On the lossless-join decomposition of relation scheme(s) in a fuzzy relational data model”, in: Proceedings of ISUMA 93 (Bilal M. Ayyub, ed.) IEEE Computer Society Pres, Maryland, 1993, 440–446.

    Google Scholar 

  67. G. Chen, E.E. Kerre, J. Vandenbulcke, “A computational algorithm for the FFD transitive closure and a complete axiomatization of fuzzy functional dependency”, Int. J. Intelligent Systems, 9 (1994) 421–439.

    Article  Google Scholar 

  68. C. Berge, Espaces Topologiques et Fonctions Multivoques, T. Duno, Paris, 1966.

    Google Scholar 

  69. K.K. Azad, “On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity”, J. Math. Anal. Appl. 82 (1981), 14–32.

    Article  MathSciNet  MATH  Google Scholar 

  70. S.P. Arya, M.P. Bhamini, “A note on semi US spaces”, Ranchi Univ. Math. J. 13 (1982), 60–67.

    MathSciNet  MATH  Google Scholar 

  71. S. Saha, “Fuzzy δ-continuous mappings”, J. Math. Anal. Appl. 126 (1987), 130–142.

    Article  MathSciNet  MATH  Google Scholar 

  72. S. Malakar, “On fuzzy semi-irresolute and strongly irresolute mappings”. Private communication.

    Google Scholar 

  73. M.N. Mukherjee, S.P. Sinha, “Irresolute and almost open functions between fuzzy topological spaces”, Fuzzy Sets and Systems 29 (1989), 381–388.

    Article  MathSciNet  MATH  Google Scholar 

  74. A. Kandil, E.E. Kerre, M. El-Shaffi, A. Noun, “Fuzzy strongly θ-continuous mappings”, in: Proceedings of Assiut First International Conference on Mathematics and Statistics, Part VIII (E.K. Al-Hussaini, A.M. Abou-El-Ela, eds.), Assiut, Egypt, 1990, 97–113.

    Google Scholar 

  75. M.N. Mukherjee, B. Ghosh, “Some stronger forms of fuzzy continuous mappings on fuzzy topological spaces”, Fuzzy Sets and Systems 38 (1990), 375–387.

    Article  MathSciNet  MATH  Google Scholar 

  76. S.P. Sinha, “On S*-closedness in fuzzy setting”. Private communication.

    Google Scholar 

  77. I. Hanafy, “A study on non-continuous functions in fuzzy topological spaces and good extensions of R 1-ness and paracompactness”, PhD thesis, Tanta, Egypt, 1990.

    Google Scholar 

  78. M.E. Abd El-Monsef, I.M. Hanafy, S.N. El-Deeb, “Semi strongly θ-continuous functions”, Indian J. Pure Appl. Math. 28(8) (1990), 721–728.

    MathSciNet  Google Scholar 

  79. T. Noiri, “On θ-semicontinuous functions”, Indian J. Pure Appl. Math. 21(5) (1990), 410–415.

    MathSciNet  MATH  Google Scholar 

  80. E.E. Kerre, N. Morsi, “On co-optimal lifts in the category of fuzzy neighbourhood spaces”, in: Proceedings IFSA 91 (B. Lowen, M. Roubens, eds.) Brussels, 1991,105–108.

    Google Scholar 

  81. A. Kandil. E.E. Kerre. A. Nouh, “Operations and mappings on fuzzy topological spaces”, Annales de la Societe Scientiflque de Bruxelles, T.105, 4 (1991), 167–188.

    MathSciNet  Google Scholar 

  82. E.E. Kerre, A. Nouh, A. Kandil, “Operations on the class of fuzzy sets on a universe endowed with a fuzzy topology”, J. Math. Anal. Appl. 180(2) (1993), 325–341.

    Article  MathSciNet  MATH  Google Scholar 

  83. A. Kandil, E.E. Kerre, M.E. El-Shafee, A. Nouh, Quasi “θ-spaces and pairwise x30B8:-perfect irreducible mappings”, J. Austral. Math. Soc. (Series A) 52 (1992), 322–333.

    Article  MATH  Google Scholar 

  84. A. Kandil, E.E. Kerre, A. Noun, M El-Shafee, “Generalized mappings between fuzzy topological spaces”, Mathematica Pannonica 3/2 (1992), 59–71.

    Google Scholar 

  85. A. Kandil, E.E. Kerre, A. Nouh, M. El-Shafee, “Fuzzy θ-perfect irreducible mappings and fuzzy θ-proximity spaces”, Fuzzy Sets and Systems 45 (1992), 93–102.

    Article  MathSciNet  MATH  Google Scholar 

  86. E.E. Kerre, A. Nouh, A. Kandil, “Generalized compactness in fuzzy topological spaces”, Matematicki Vesnik 43 (1991), 29–40.

    MathSciNet  MATH  Google Scholar 

  87. M.K. El Gayyar, E.E. Kerre, A.A. Ramadan, “Almost compactness and near compactness in smooth topological spaces”, Fuzzy Sets and Systems 62 (1994), 193–202.

    Article  MathSciNet  MATH  Google Scholar 

  88. M.K. El Gayyar, E.E. Kerre, A.A. Ramadan, “Separation axioms in smooth topological spaces”, submitted to Far-East Journal of Mathematical Sciences (1993).

    Google Scholar 

  89. B. De Baets, E.E. Kerre, “Fuzzy subcontinuity”, in: Proceedings of NAFIPS’ 93, (M. McAllister, ed.) Allentown, P, USA 1993, 232–236.

    Google Scholar 

  90. B. De Baets, E.E. Kerre, “On the supercomposition of subcontinuous and fuzzy subcontinuous mappings and related topics”, Journal of Fuzzy Mathematics 2 (1994), 17–34.

    MathSciNet  MATH  Google Scholar 

  91. P. Smets, P. Magrez, “Implication in fuzzy logic”, Int. J. Approximate Reasoning 1 (1987), 327–347.

    Article  MathSciNet  MATH  Google Scholar 

  92. B. Cappelle, E.E. Kerre, D. Ruan, F. Vanmassenhove, “Characterization of the binary operations on the unit interval satisfying the generalized modus ponens inference rule”, Mathematica Pannonica 2/1 (1991), 105–121.

    Google Scholar 

  93. E.E. Kerre, “A comparative study of the behavior of some popular fuzzy implication operators on the generalized modus ponens”, in: Fuzzy Logic for the Management of Uncertainty (L. Zadeh, J. Kacprzyck, eds.) J. Wiley and Sons, New York, 1992, 281–296.

    Google Scholar 

  94. D. Ruan, E.E. Kerre, G. De Cooman, B. Cappelle, F. Vanmassenhove, “Influence of the fuzzy implication operator on the method-of-cases inference rule”, in: Int. J. Approx. Reas. vol.4, no.4 (1990), 307–318.

    Article  MATH  Google Scholar 

  95. M. Mizumoto, H. Zimmerman, “Comparison of fuzzy reasoning”, Fuzzy Sets and Systems 8 (1982), 253–283.

    Article  MathSciNet  MATH  Google Scholar 

  96. D. Ruan, E.E. Kerre, “On the extension of the compositional rule of inference”, Int. J. Intell. Syst. 8 (1993), 807–817.

    Article  MATH  Google Scholar 

  97. D. Ruan, E.E. Kerre, “Fuzzy implication operators and generalized fuzzy method of cases”, Fuzzy Sets and Systems 54 (1993), 23–37.

    Article  MathSciNet  MATH  Google Scholar 

  98. A. Van Schooten, E. Kerre, R. De Caluwe, “An expert system development tool using fuzzy logic” in: Proceedings Intelligent Autonomous Systems (L. Hertzberger, ed.) Elsevie, Amsterdam, 1986, pp. 668–677.

    Google Scholar 

  99. A. Van Schooten, E. Kerre, R. De Caluwe, “Fuzzy expert systems”, in: Proceedings The Intelligent Access to Information (R. De Caluwe, ed.) FBVI, Brussels, 1987, 245–257.

    Google Scholar 

  100. A. Van Schooten, E. Kerre, R. De Caluwe, “Approximate reasoning based on possibility theory”, in: Cybernetics and Systems (R. Trappl, ed.) Kluwer Academic Publisher, Dordrecht, 1988, 643–650.

    Google Scholar 

  101. G. De Cooman, E.E. Kerre, B. Cappelle, D. Ruan, F. Vanmassenhove, “On the extension of classical propositional logic by means of a triangular norm”, Int. J. Intell. Syst. 5, no; 3 (1990), 307–322.

    Article  MATH  Google Scholar 

  102. B. De Baets, E.E. Kerre, S. Nasuto, “Measuring fuzziness: an evaluation problem”, in Proceedings ISUMA 93 (B. Ayyub, ed.) Maryland, 1993,65–72.

    Google Scholar 

  103. B. De Baets, S. Nasuto, E.E. Kerre, “An order — theoretic approach to the measuring of fuzziness”, in: Uncertainty Modeling and Analysis: Theory and Applications (B. Ayyub, M.M. Gupta, eds.), Elsevier Science Publisher, Amsterdam, 1994,173–191.

    Google Scholar 

  104. R. Barlow, S. Wu, “Coherent systems with multistate components”, Mathematics of Operations Research, 3 (1978), 275–281.

    Article  MathSciNet  MATH  Google Scholar 

  105. B. Cappelle, E.E. Kerre, F. Vanmassenhove, “A possibilistic uncertainty model applied to the Barlow-Wu class of multistate structures”, in: Proceedings IFSA 91 (B. Lowen, M. Roubens, eds.) Brussels, 1991, 25–28.

    Google Scholar 

  106. B. Cappelle, E.E. Kerre, “On a possibilistic approach to reliability theory”, in: Proceedings ISUMA 93 (B. Ayyub, ed.) Maryland, 1993, 415–418.

    Google Scholar 

  107. J. Montero, B. Cappelle, E.E. Kerre, “The usefulness of complete lattices in reliability theory”, in: Fuzzy Sets and Possibilistic Theory in Reliability and Safety Analysis (J. Kacprzyck, T. Onisawa, eds.), Omnitech Press, Warsaw and Physica-Verla, Heidelberg, 1993, in press.

    Google Scholar 

  108. B. Cappelle, E.E. Kerre, “Issues on possibilistic reliability theory”, in: Fuzzy Sets and Possibilistic Theory in Reliability and Safety Analysis (J Kacprzyck, T. Onisawa, eds.), Omnitech Press, Warsaw and PhysicaVerla, Heidelberg, 1993, in press.

    Google Scholar 

  109. B. Cappelle, E.E. Kerre, “Computer assisted reliability analysis: an application of possibilistic reliability theory to a subsystem of a nuclear power plant”, in: Fuzzy Logic and Intelligent Technologies in Nuclear Science (D. Ruan, P. D’hondt, P. Govaerts, E.E. Kerre, eds.) World Scientific Publishing Compan, Singapore, 1994, 260–265.

    Google Scholar 

  110. B. Cappelle, E.E. Kerre, “The CARA system: an intelligent tool to support qualitative reliability analysis”, in: Proceedings International AMSE Conference on Systems, Analysis, Control & Design, 1 (1994), 25–35.

    Google Scholar 

  111. B. Cappelle, E.E. Kerre, “A general possibilistic framework for reliability theory”, in: Proceedings 5th International Conference Information Processing and Managament of Uncertainty in Knowledge-Based Systems 1 (1994), 832–837.

    Google Scholar 

  112. B. Cappelle, E.E. Kerre, “Possibilistic and necessistic reliability functions: fundamental concepts and theorems to represent non-probabilistic uncertainty in reliability theory”, in: Uncertainty Modeling and Analysis: Theory and Applications (B. Ayyub, M.M. Gupta, eds.) Elsevier Science Publisher, Amsterdam, 1994,131–144.

    Google Scholar 

  113. F. Suarez Garcia, P. Gil Alvarez, “Two families of fuzzy integrals”, Fuzzy Sets and Systems 18 (1986), 67–81.

    Article  MathSciNet  MATH  Google Scholar 

  114. G. De Cooman, E.E. Kerre, B. Cappelle, D. Ruan, F. Vanmassenhove, “On the extension of classical propositional logic by means of a triangular norm”, in:Proceedings Third IFSA Congress, Seattle, 1989, 821–824.

    Google Scholar 

  115. G. De Cooman, E.E. Kerre, F. Vanmassenhove, “Possibility theory: an integral theoretic approach”, Fuzzy Sets and Systems 46 (1992), 287–299.

    Article  MathSciNet  MATH  Google Scholar 

  116. G. De Cooman, E.E. Kerre, “Order norms on partially ordered sets”, in: Proceedings Conference Intellectual Systems, Moscow, 1993.

    Google Scholar 

  117. G. De Cooman, E.E. Kerre, “Ample fields”, Simon Stevin 67 (1993), 235–244.

    MathSciNet  MATH  Google Scholar 

  118. G. De Cooman, E.E. Kerre, “Order norms on bounded partially ordered sets”, Journal of Fuzzy Mathematics 2 (1994), 281–310.

    MathSciNet  MATH  Google Scholar 

  119. G. De Cooman, E.E. Kerre, “A new approach to possibilistic independence”, in: Proceedings 3rd. IEEE International Conference on Fuzzy Systems, 1994, 1446–1451.

    Google Scholar 

  120. E.E. Kerre, B. Raman, “Application of fuzzy programming to ship steering”, in: Approximate Reasoning in Expert Systems, (M. Gupta, A. Kandel, W. Bandler, J. Kiszka, eds.) North-Hollan, Amsterdam, 1985, 719–730.

    Google Scholar 

  121. E.E. Kerre, “Outline of an expert system for ECG diagnosis using fuzzy sets”, Artificial Intelligence in Medicine 1 (1989), 139–144.

    Article  Google Scholar 

  122. E.E. Kerre, “A call for crispness in fuzzy set theory”, Fuzzy Sets and Systems 29 (1989), 57–65.

    Article  MathSciNet  MATH  Google Scholar 

  123. E.E. Kerre, “Fuzzy set theory in information retrieval and databases”, in: Encyclopedia of Computer Science and Technology (A. Kent, J. Williams, eds.) Marcel Dekker Inc., New York, vol 21, Suppl.6, 1990, 117–138.

    Google Scholar 

  124. P. Poulosky, D. Holliday, J. Niemann, E.E. Kerre, “Calculus of α-level fuzzy sets”, Report Series #115, University of Nebrask, Lincoln, U.S.A. (1990).

    Google Scholar 

  125. D. Guinan, K. Streicher, E.E. Kerre, “Set-theoretic properties of the class of fuzzy sets endowed with the bounded sum and the bold intersection”, Report Series #114, University of Nebraska, Lincoln,U.S.A. (1990).

    Google Scholar 

  126. E.E. Kerre, “The suitability of fuzzy set theory for the representation and manipulation of imprecision and uncertainty in knowledge based systems”, Communication & Cognition, vol.8, no. 1 (1991), 49–79.

    MathSciNet  Google Scholar 

  127. E.E. Kerre, “Fuzzy logic and medical diagnosis”, in: Proceedings of the Symposium Clear Applications of Fuzzy Logic, Delft, 1991, 160–174.

    Google Scholar 

  128. B. Cappelle, G. De Cooman, E.E. Kerre, F. Vanmassenhove, “Intelligent reductions methods based upon fuzzy set theory as a helpful tool for controlling industrial plants”, Fuzzy Sets and Systems 50, (1992), 119–126.

    Article  Google Scholar 

  129. E.E. Kerre, “Operations on fuzzy sets”, in: Vage verzamelingen en vage logica (J. De Kerf, ed.) Technologisch Instituut KVIV, Antwerpen, 1992,1–27.

    Google Scholar 

  130. B. De Baets, E.E. Kerre, “Kite-tail lattices and their characterization”, Far-East Journal of Mathematical Sciences, vol. 1 (1993), 1–11.

    MathSciNet  MATH  Google Scholar 

  131. B. De Baets, E.E. Kerre, “The generalized modus ponens and the triangular fuzzy data model”, Fuzzy Sets and Systems, Special Issue on Fuzzy Data Analysis (K. Hirota, ed.), 59 (1993), 305–317.

    Google Scholar 

  132. B. De Baets, E.E. Kerre, “An introduction to fuzzy mathematical morphology”, in: Proceedings NAFIPS 93 (Mc-Allister, ed.) Allentown, U.S.A., 1993, 129–133.

    Google Scholar 

  133. B. De Baets, E.E. Kerre, “Fuzzy relational compositions”, Fuzzy Sets and Systems 60 (1993), 109–120.

    Article  MathSciNet  MATH  Google Scholar 

  134. B. De Baets, E.E. Kerre, “A primer on solving fuzzy relational equations on the unit interval”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2 (1994), 205–225.

    Article  MathSciNet  MATH  Google Scholar 

  135. B. Van de Walle, D. Ruan, E.E. Kerre, “Applications of fuzzy reasoning in nuclear decision aiding systems”, Fuzzy Systems and Artificial Intelligence, vol. II, no.2 (1993).

    Google Scholar 

  136. E. Tsiporkova-Hristoskova, E.E. Kerre, “Pure and stable fuzzy sets”, in: Proceedings International AMSE Conference on Systems, Analysis, Control & Design, vol. 1,1994, 3–12.

    Google Scholar 

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Authors and Affiliations

  1. Fuzziness and Uncertainty Modeling, Universiteit Gent, Krijgslaan 281-S9, B-9000, Gent, Belgium

    Etienne Kerre

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  1. Etienne Kerre
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  1. Belgian Nuclear Research Centre (SCK · CEN), Mol, Belgium

    Da Ruan

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Kerre, E. (1995). On the Evolution of the Mathematics of Fuzziness. In: Ruan, D. (eds) Fuzzy Set Theory and Advanced Mathematical Applications. International Series in Intelligent Technologies, vol 4. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2357-4_1

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  • DOI: https://doi.org/10.1007/978-1-4615-2357-4_1

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