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Mathematics and Fuzziness

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Fuzzy Sets Theory and Applications

Part of the book series: NATO ASI Series ((ASIC,volume 177))

Abstract

The purpose of this work is to introduce the reader to two aspects of fuzzy sets and mathematics.

The logical aspect where we present some theories to wellfound fuzzy sets and its basic concepts as they are known now and secondly and most importantly, the analytical aspect where we introduce a new and more restricted theory of fuzziness.

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References

  1. C. Alsina, E. Trillas, L. Valverde, ‘On some logical connectives for fuzzy set theory’, J. Math. Anal. Appl. 93, 15–26 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. J.M. Anthony, H. Sherwood, ‘Fuzzy groups redefined’, J. Math. Anal. Appl. 69, 124–130 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Artico, R. Moresco, ‘Fuzzy proximities and totally bounded fuzzy uniformities’, J. Math. Anal. Appl. 99, 320–337 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Badard, ‘Fuzzy pretopological spaces and their representation’, J. Math. Anal. Appl. 81, 378–390 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Bellman, M. Giertz, ‘On the analytic formalism of the theory of of fuzzy sets’, Information Sci. 5, 149–156 (1973)

    Article  MathSciNet  Google Scholar 

  6. J.C. Bezdek, B. Spillman, R. Spillman, ‘A fuzzy relation space for group decision theory’, Fuzzy Sets and Systems 1, 255–263 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. J.C. Bezdek, B. Spillman, R. Spillman, ‘Fuzzy relational spaces for group decision theory: An application’, Fuzzy Sets and Systems 2, 5–14 (1979)

    Article  MATH  Google Scholar 

  8. J.A. BÜLBÖL, ‘On connections between the countability properties of (X,w(T)) and (X,T)’, Fuzzy Sets and Systems 15, 91–98 (1985)

    Article  MathSciNet  Google Scholar 

  9. L. Caire, U. Cerruti, ‘Fuzzy relational spaces and algebraic structures’, Rend. Sem. Mat. Univers. Politeoh. Torino 35, 443–461 (1976)

    MathSciNet  Google Scholar 

  10. U. Cerruti, ‘The Stone-Cech compactification in the category of fuzzy topological spaces’, Fuzzy Sets and Systems 6, 197–204 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. U. Cerruti, ‘Completion of L-fuzzy relations’, J. Math. Anal. Appl. 94, 312–327 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. U. Cerruti, U. HÖhle, ‘Categorical foundations of probabilistic microgeometry’, preprint

    Google Scholar 

  13. J.L. Coulon, J. Coulon, ‘Fuzzy Boolean algebras’, J. Math. Anal. Appl. 99, 248–256 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. C. De Mitri, E. Pascali, ‘Characterization of fuzzy topologies from neigborhoods of fuzzy points’, J. Math. Anal. Appl. 93, 1–14 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Dubois, H. Prade, ‘Operations on fuzzy numbers’, Internat. J. Systems Sci. 9(6), 613–626 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  16. M.A. Erceg, ‘Functions, equivalence relations, quotient spaces and subsets in fuzzy set theory’, Fuzzy Sets and Systems 3, 75–92 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  17. A.O. Esogbue, R.C. Elder, ‘Fuzzy Sets and the modelling of physician decision processes. Part I: the initial interview-information gathering session’, Fuzzy Sets and Systems 2, 279–292 (1979)

    Article  MATH  Google Scholar 

  18. A.O. Esogbue, R.C. Eler, ‘Fuzzy Sets and the modelling of physician decision processes. Part II: Fuzzy diagnosis decision models’, Fuzzy Sets and Systems 3, 1–10 (1980)

    Article  MATH  Google Scholar 

  19. D.H. Foster, ‘Fuzzy topological groups’, J. Math. Anal. Appl. 67, 549–564 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. M.J. Frank, ‘Probabilistic topological spaces’, J. Math. Anal. Appl. 34, 67–81 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  21. T.E. Gantner, R.C. Steinlage, R.H. Warren, ‘Compactness in fuzzy topological spaces’, J. Math. Anal. Appl. 62, 547–562 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Goetschel Jr., W. Voxman, ‘A pseudometric for fuzzy sets and certain related results’, J. Math. Anal. Appl. 81, 507–523 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  23. J.A. Goguen, ‘L-fuzzy sets’, J. Math. Anal. Appl. 18, 145–174 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Goldblatt, Topoi, the categorical analysis of logic, North Holland Publishing CY (1979)

    Google Scholar 

  25. S. Gottwald, ‘Untersuchungen zur mehrwertigen Mengenlehre’, Parts I, II, II, Math. Nachr. 72, 297–303 (1976); 74, 329-336 (1976); 79, 207-217 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  26. D. Higgs, ‘A category approach to Boolean valued set theory’, preprint

    Google Scholar 

  27. U. HÖhle,‘Probabilische Topologiën’, Manuscrip ta Math. 26, 223–245 (1978)

    Google Scholar 

  28. U. HÖhle, ‘Probabilistisch kompakte L-unscharfe mengen’, Manuscripta Math. 26, 331–347 (1979)

    Article  MATH  Google Scholar 

  29. U. HÖhle, ‘Probabilistic uniformization of fuzzy topologies’, Fuzzy Sets and Systems 1, 311–332 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. U. HÖhle, ‘Upper semicontinuous fuzzy sets and applications’, J. Math. Anal. Appl. 78, 659–673 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  31. U. HÖhle, ‘Fuzzy topologies and topological space objects in a topos’, Busefal 21, 4–10 (1985)

    Google Scholar 

  32. B. Hutton, ‘Normality in fuzzy topological spaces’, J. Math. Anal. Appl. 50, 74–79 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  33. B. Hutton, ‘Uniformities on fuzzy topological spaces’, J. Math. Anal. Appl. 58, 559–571 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  34. B. Hutton, I. Reilly, ‘Separation axioms in fuzzy topological spaces’, Fuzzy Sets and Systems 3, 93–104 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  35. A.K. Katsaras, D.B. Liu, ‘Fuzzy vectorspaces and fuzzy topological spaces’, J. Math. Anal. Appl. 58, 135–146 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  36. A.K. Katsaras, ‘Ordered fuzzy topological spaces’, J. Math. Anal. Appl. 84, 44–58 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  37. A.K. Katsaras, ‘Fuzzy topological vectorspaces I’, Fuzzy Sets and Systems 6, 85–96 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  38. A.K. Katsaras, ‘Fuzzy topological vectorspaces II’, Fuzzy Sets and Systems 12, 143–154 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. A.K. Katsaras, ‘Linear fuzzy neighborhood spaces’, Fuzzy Sets and Systems 16, 25–40 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  40. A.J. Klein, ‘α-closure in fuzzy topology’, Rooky Mount. J. Math. 11, 553–560 (1981)

    Article  MATH  Google Scholar 

  41. A.J. Klein, ‘Generating fuzzy topologies with semiclosure operations’, Fuzzy Sets and Systems 9, 267–274 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  42. A.J. Klein, ‘Generalizing the L-fuzzy unit interval’, Fuzzy Sets and Systems 12, 271–280 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  43. A.J. Klein, ‘Fuzzy topologies with specified level properties’, preprint

    Google Scholar 

  44. A.J. Klein, ‘When is a fuzzy topology topological ?’, preprint

    Google Scholar 

  45. E.P. Klement, ‘Fuzzy σ-algebras and fuzzy measurable functions’, Fuzzy Sets and Systems 4, 83–94 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  46. E.P. Klement, ‘Characterization of finite fuzzy measures using Markoff-kernels’, J. Math. Anal. Appl. 75, 330–339 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  47. E.P. Klement, R. LoWen, W. Schwyhla, ‘Fuzzy probability measures’, Fuzzy Sets and Systems 5, 21–30 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  48. E.P. Klement, W. Schwyhla, ‘Correspondence between fuzzy measures and classical measures’, Fuzzy Sets and Systems 7, 57–70 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  49. E.P. Klement, A theory of fuzzy measures: a survey, Fuzzy Information and Decision Processes, M.M. Gupta, E. Sanchez (eds.), North Holland Publishing CY (1982)

    Google Scholar 

  50. E.P. Klement, ‘Operations on fuzzy sets — an axiomatic approach’, Inf. Sci. 27, 221–232 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  51. E.P. Klement, ‘Some remarks on a paper by R.R. Yager’, Inf. Sci. 27, 211–220 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  52. E.P. Klement, ‘Construction of fuzzy σ-algebras using triangular norms’, J. Math. Anal. Appl. 85, 543–565 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  53. E.P. Klement, ‘Fuzzy measures assuming their values in the set of fuzzy numbers’, J. Math. Anal. Appl. 93, 312–323 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  54. E. Lowen, R. Lowen, ‘Characterization of convergence in fuzzy topological spaces’, Int. J. Math, and Math. Sci., in print

    Google Scholar 

  55. E. Lowen, R. Lowen, ‘On measures of compactness in fuzzy topological spaces’, J. Math. Anal. Appl., in print

    Google Scholar 

  56. R. Lowen, ‘On fuzzy complements’, Inf. Sci. 14, 107–113 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  57. R. Lowen, ‘Convergence in fuzzy topological spaces’, General Topology Appl. 10, 147–160 (1979)

    Article  MathSciNet  Google Scholar 

  58. R. Lowen, ‘Convex fuzzy sets’, Fuzzy Sets and Systems 3, 291–310 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  59. R. Lowen, ‘Compact Hausdorff fuzzy topological spaces are topological’, Topology Appl. 12, 65–74 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  60. R. Lowen, ‘Fuzzy uniform spaces’, J. Math. Anal. Appl. 82, 370–385 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  61. R. Lowen, ‘Connectedness in fuzzy topological spaces’, Rooky Mt. J. Math. 11, 427–433 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  62. R. Lowen, ‘Fuzzy neighborhood spaces’, Fuzzy Sets and Systems 7, 165–189 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  63. R. Lowen, P. Wuyts, ‘On completeness, compactness and precompactness in fuzzy uniform spaces’, Part I, J. Math. Anal. Appl. 90, 563–581 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  64. R. Lowen, P. Wuyts, ‘On completeness, compactness and precompactness in fuzzy uniform spaces’, Part II, J. Math. Anal. Appl. 92, 342–371 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  65. R. Lowen, ‘Ix, the hyperspace of fuzzy sets, a natural non topological fuzzy topological space’, Trans. Amer. Math. Soo. 278, 547–564 (1983)

    MathSciNet  MATH  Google Scholar 

  66. R. Lowen, ‘Hyperspaces of fuzzy sets’, Fuzzy Sets and Systems 9, 287–311 (1983)

    Article  MathSciNet  Google Scholar 

  67. R. Lowen, ‘On the existence of natural fuzzy topologies on spaces of probability measures’, Math. Nachr. 115, 33–57 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  68. R. Lowen, ‘Fuzzy convergence versus weak convergence in spaces of probability measures’, Abhandelungen Akad. Wiss. DDR 2, 147–152 (1984)

    MathSciNet  Google Scholar 

  69. R. Lowen, ‘On the fundamentals of fuzzy sets’, Stochastica 8, 157–169 (1984)

    MathSciNet  MATH  Google Scholar 

  70. R. Lowen, ‘On the relation between fuzzy equalities, pseudometrics and uniformities’, Quaestiones Math. 7, 407–419 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  71. R. Lowen, ‘The order aspect of the fuzzy real line’, Manuscripta Math. 39, 293–309 (1985)

    Article  MathSciNet  Google Scholar 

  72. R. Lowen, ‘Metric spaces viewed as fuzzy topological spaces induced by Lipschitz functions’, Math. Nachr. 120, 249–265 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  73. R. Lowen, P. Wuyts, ‘Concerning the constants in fuzzy topology’, J. Math. Anal. Appl., in print

    Google Scholar 

  74. R. Lowen, On the existence of natural non-topological fuzzy topological spaces, Research and Exposition in Mathematics 11, Heldermann Verlag (1985)

    Google Scholar 

  75. R. Lowen, ‘A complete characterization of the levelspaces of IR( I) and I(I), Rocky Mt. J. Math., in print

    Google Scholar 

  76. R. Lowen, ‘Mathematics and fuzziness: some personal reflections’, Inf. Sci., in print

    Google Scholar 

  77. R. Lowen, P. Maes, ‘Completeness in hyperspaces of compact fuzzy sets’, submitted for publication

    Google Scholar 

  78. R. Lowen, ‘Approach spaces’, submitted for publication

    Google Scholar 

  79. E.G. Manes, ‘Aclass of fuzzy theories’, J. Math. Anal. Appl. 85, 409–451 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  80. H.W. Martin, ‘A Stone-Cech ultra fuzzy compactification’, J. Math. Anal. Appl. 73, 453–456 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  81. H.W. Martin, ‘Weakly induced fuzzy topological spaces’, J. Math. Anal. Appl. 78, 634–639 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  82. S.V. Ovchinnikov, ‘Structure of fuzzy binary relations’, Fuzzy Sets and Systems 6, 169–197 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  83. H. PoincarÉ, Science and method, translated by F. Maitland, Dover publications

    Google Scholar 

  84. P-M. Pu, Y-M. Liu, ‘Fuzzy topology I. Neighborhood structure of a fuzzy point and Moore Smith convergence’, J. Math. Anal. Appl. 76, 571–599 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  85. P-M. Pu, Y-M. Liu, ‘Fuzzy topology II. Product and quotient spaces’, J. Math. Anal. Appl. 77, 20–37 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  86. M.L. Puri, D.A. Ralescu, ‘Differentials of fuzzy functions’, J. Math. Anal. Appl. 91, 552–558 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  87. D.A. Ralescu, ‘Toward a general theory of fuzzy variables’, J. Math. Anal. Appl. 86, 176–193 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  88. S.E. Rodabaugh, ‘Suitability in fuzzy topological spaces’, J. Math. Anal. Appl. 79, 273–285 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  89. S.E. Rodabaugh, The L-fuzzy real line and its subspaces, Recent developments in Fuzzy Set and Possibility Theory, Pergamon Press (1982)

    Google Scholar 

  90. S.E. Rodabaugh, ‘A categorical accomodation of various notions of fuzzy topology’, Fuzzy Sets and Systems 9, 241–266 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  91. S.E. Rodabaugh, ‘A lattice of continuities for fuzzy topological spaces’, J. Math. Anal. Appl. 79, 244–255 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  92. S.E. Rodabaugh, ‘Fuzzy addition in the L-fuzzy real line’, Fuzzy Sets and Systems 8, 39–52 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  93. S.E. Rodabaugh, ‘Separation axioms and the fuzzy real line’, Fuzzy Sets and Systems, 163–154 (1984)

    Google Scholar 

  94. S.E. Rodabaugh, ‘Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines’, Fuzzy Sets and Systems 15, 285–310 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  95. R. Srivastava, S.N. Lal, A.K. Srivastava, ‘Fuzzy Hausdorff topological spaces, J. Math. Anal. Appl. 81, 497–506 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  96. A.K. Srivastava, R. Srivastava, ‘Fuzzy Sierpinski Space: another note’, Fuzzy Mathematics 5, 99–104 (1985)

    MathSciNet  MATH  Google Scholar 

  97. E. Trillas, ‘Sobre funciones de Negación en la Téoría de Conjuntos Difusos’, Stochastica 3, 47–59 (1979)

    MathSciNet  MATH  Google Scholar 

  98. G-J. Wang, ‘A new fuzzy compactness defined by fuzzy nets’, J. Math. Anal. Appl. 94, 1–23 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  99. R.H. Warren, ‘Fuzzy topologies characterized by neighborhood systems’, Rocky Mt. J. Math. 9(4), 761–764 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  100. S. Weber, ‘A general concept of fuzzy connectives negations and implications based on t-norms and co-t-norms’, Fuzzy Sets and Systems 11, 115–134 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  101. S. Weber, ‘Measures of fuzzy sets and measure of fuzziness’, Fuzzy Sets and Systems 13, 247–272 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  102. M.D. Weiss, ‘Fixed points, separation and induced topologies for fuzzy sets’, J. Math. Anal. Appl. 50, 142–150 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  103. P. Wuyts, R. Lowen, ‘On separation axioms in fuzzy topological spaces, fuzzy neighborhood spaces and fuzzy uniform spaces’, J. Math. Anal. Appl. 93, 27–41 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  104. P. Wuyts, ‘On the determination of fuzzy topological spaces and fuzzy neighborhood spaces by their level topologies’, Fuzzy Sets and Systems 12, 71–85 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  105. P. Wuyts, R. Lowen, ‘On local and global measures of separation in fuzzy topological spaces’, Fuzzy Sets and Systems, in print

    Google Scholar 

  106. L.A. Zadeh, ‘Fuzzy Sets’, Information and Control 8, 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  107. L.A. Zadeh, ‘Fuzzy Sets as a basis for a theory of possibility’, Fuzzy Sets and Systems 1, 3–29 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  108. L.A. Zadeh, Making computers think like people, I.E.E.E. Spectrum, August 1984

    Google Scholar 

  109. H.J. Zimmermann, ‘Description and optimization of fuzzy systems’, Int. J. General Syst. 2(4) (1975)

    Google Scholar 

  110. H.J. Zimmermann, ‘Fuzzy programming and linear programming with several objective functions’, Fuzzy Sets and Systems 1, 45–56 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dienst Wiskundige Analyse, University of Antwerp R.U.C.A., Groenenborgerlaan 171, Antwerpen, 2020, Belgium

    R. Lowen

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  1. R. Lowen
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Editors and Affiliations

  1. Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    André Jones

  2. O.R. and M.S Consultant, Grenoble, France

    Arnold Kaufmann

  3. Institute of Technology, Aachen, Germany

    Hans-Jürgen Zimmermann

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© 1986 D. Reidel Publishing Company

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Lowen, R. (1986). Mathematics and Fuzziness. In: Jones, A., Kaufmann, A., Zimmermann, HJ. (eds) Fuzzy Sets Theory and Applications. NATO ASI Series, vol 177. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4682-8_1

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  • DOI: https://doi.org/10.1007/978-94-009-4682-8_1

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