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On Some Fuzzy Categories Related To Category L-TOP Of L-Topological Spaces

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Topological and Algebraic Structures in Fuzzy Sets

Part of the book series: Trends in Logic ((TREN,volume 20))

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Abstract

Since the inception of fuzzy set theory (see [73], [51], [16]), the efforts of many researchers were directed to find fuzzy analogues of basic concepts of classical mathematics and to develop corresponding theories. General topology was one of the first fields to which the interest of “fuzzy mathematicians” was focused. Already as soon as in 1968 topological structures on lattices of [0,1]-fuzzy sets were considered by C. L. Chang [5]; C. L. Chang’s ideas for more general lattices were developed by J.A. Goguen [17], B. Hutton [34], [35], S. E. Rodabaugh [46], [48], U. Höhle [21], P. Eklund [9], [10], P. Eklund and W. Gähler [11] et al. However in all these works the fuzziness revealed itself only on the level of the powerset, while the topological structure still remained crisp. In particular, in the Chang-Goguen approach an L-fuzzy topology on set X is a family τ ⊂ L X satisfying natural analogues of the usual axioms of topology. In the sequel, following [30], such family τ will be referred to as an L-topology on a set X, see 3.1.

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References

  1. R. Artico, G. Moresco, Fuzzy proximities according with Lowen fuzzy uniformities, Fuzzy Sets and Systems 21 (1987), 85–98.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloquium Publication XXXV, 3rd ed., American Mathematical Society (Providence, RI ), 1967.

    Google Scholar 

  3. M. Burgin, A. Sostak, Towards the theory of continuity defect and continuity measure for mappings of metric spaces, Acta Univ. Latv. 576 (1992), 45–62.

    MathSciNet  Google Scholar 

  4. M. Burgin, A. Sostak, Fuzzification of the theory of continuous functions, Fuzzy Sets and Systems 62 (1994), 71–81.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), pp. 182–190.

    Article  MathSciNet  MATH  Google Scholar 

  6. E. C. Topological Spaces, English translation, Interscience-Wiley (London-New York ), 1966.

    Google Scholar 

  7. J. Coulon, J. L. Coulon, Une notion de proximite floue, Busefal 10 (1990), 17–34.

    Google Scholar 

  8. D. Dzhajanbajev, A. Sostak, On a fuzzy uniform structure, A.ta et Comm. Univ. Tartu. 940 (1992), 31–36.

    MathSciNet  Google Scholar 

  9. P. Eklund, Category theoretic properties of fuzzy topological spaces, Fuzzy Sets and Systems 19 (1984), 303–310.

    Google Scholar 

  10. P. Eklund, Categorical fuzzy topology, Acta Acad. Aboensis Ser. B 46: 1 (1986), 1–13.

    Google Scholar 

  11. P. Eklund, W. Gähler, Basic notions for fuzzy topology, I, Fuzzy Sets and Systems 26 (1988), 333–356.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. W. Gähler, A. S. Abd-Allah, A. Kandil, On extended fuzzy topologies, Fuzzy Sets and Systems 109 (2000), 149–172.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. H. Ghanim, H. M. H.san, L-closure spaces, Fuzzy Sets and Systems 33 (1989), 383–391.

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott, A Compendium Of Continuous Lattices, Springer Verlag (Berlin/Heidelberg/New York), 1980.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. J. A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl. 43 (1973) 737–742.

    Google Scholar 

  18. R. Goldblatt, Topoi: The Categorical Analysis Of Logic, revised ed., North-Holland (Amsterdam ), 1981.

    Google Scholar 

  19. P. C. Hammer, Extended topology: set-valued set-functions, Nieuw Archief voor Wiskunde 3: 10 (1962), 55–77.

    Google Scholar 

  20. H. Herrlich, G. E. Strecker, Category Theory, Heldermann Verlag (Berlin), 1979.

    Google Scholar 

  21. U. Höhle, Probabilistische topologies, Manuscripta Math. 26 (1978), 223–245.

    Article  MathSciNet  Google Scholar 

  22. U. Höhle, Upper semi continuous fuzzy sets and applications, J. Math. Anal. Appl. 78 (1980) 659–673.

    Article  MathSciNet  Google Scholar 

  23. U. Höhle, G-fuzzy topologies on algebraic structures, J. Math. Anal. Appl. 108 (1985), 113–150.

    Article  MathSciNet  Google Scholar 

  24. U. Höhle, Fuzzy topologies and topological space objects in a topos, Fuzzy Sets and Systems 19 (1986), 299–304.

    Article  MathSciNet  Google Scholar 

  25. U. Höhle, Monoidal closed categories, weak topoi and generalized logics, Fuzzy Sets and Systems 42 (1991), 15–35.

    Article  MathSciNet  Google Scholar 

  26. U. Höhle, M-valued sets and sheaves over integral commutative cl-monoids, Chapter 2 in: S. E. Rodabaugh, E. P. Klement, U. Höhle, eds, Application Of Category To Fuzzy Sets, Theory and Decision Library—Series B: Mathematical and Statistical Methods, Volume 14, Kluwer Academic Publishers (Boston/Dordrecht/London), 1992, pp. 33–72.

    Google Scholar 

  27. U. Höhle, Commutative residuated 1-monoids, Chapter IV in: U. Höhle, E. P. Klement, eds, Non-Classical Logics And Their Applications To Fuzzy Subsets—A Handbook of the Mathematical Foundations of Fuzzy Set Theory, Theory and Decision Library—Series B: Mathematical and Statistical Methods, Volume 32, Kluwer Academic Publishers (Boston/Dordrecht/London), 1995, pp. 53–106.

    Google Scholar 

  28. U. Höhle, On the foundations of fuzzy set theory, preprint, 1996.

    Google Scholar 

  29. U. Höhle, A. Sostak, A general theory of fuzzy topological spaces, Fuzzy Sets and Systems 73 (1995), 131–149.

    Article  MathSciNet  MATH  Google Scholar 

  30. U. Höhle, A. Sostak, Axiomatics of fixed-basis fuzzy topology,Chapter 3 in: U. Höhle, S. E. Rodabaugh, eds, Mathematics Of Fuzzy Sets: Logic, Topology, And Measure Theory, The Handbooks of Fuzzy Sets Series, Volume 3(1999), Kluwer Academic Publishers (Dordrecht/Boston/London), pp. 123–273.

    Google Scholar 

  31. U. Höhle, L. N. Stout, Foundations of fuzzy sets, Fuzzy Sets and Systems 40 (1991), 257–296.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  34. B. Hutton, Products of fuzzy topological spaces, Top. and Appl. 11 (1980), 59–67.

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. T. Kubiak, On Fuzzy Topologies, Doctoral Dissertation, Adam Mickiewicz University ( Poznan, Poland ), 1985 ).

    Google Scholar 

  37. T. Kubiak, The topological modification of the L-fuzzy unit interval, Chapter 11 in: S. E. Rodabaugh, E. P. Klement, U. Höhle, eds, Application Of Category To Fuzzy Sets, Theory and Decision Library—Series B: Mathematical and Statistical Methods, Volume 14, Kluwer Academic Publishers (Boston/Dordrecht/London), 1992, pp. 275–305.

    Google Scholar 

  38. T. Kubiak, A. Sostak, Lower set-valued fuzzy topologies, Quaestiones Mathematicae 20: 3 (1997), 423–429.

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu Wangling, Fuzzy invariant subgroups and fuzzy ideals,Fuzzy Sets and Systems 8 (1982), 133–139.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. R. Lowen, Initial and final fuzzy topologies and the fuzzy Tychonoff theorem, J. Math. Anal. Appl. 58 (1977), 621–633.

    Google Scholar 

  42. A. S. Mashhour, M. H. Ghanim, On closure spaces, Indian J. Pure Appl. Math. 14 (1983), 680–691.

    MathSciNet  MATH  Google Scholar 

  43. C. V. Negoita, D. A. Ralescu, Application Of Fuzzy Sets To System Analysis, John Wiley Sons (New York), 1975.

    Google Scholar 

  44. N. Netzer, Verallgemeinerte topologische Strukturen, Jahrbuch Überblicke Math. (1978), 87–196.

    Google Scholar 

  45. H. E. Porst, A. Sostak, On the concept of a fuzzy function,in preparation.

    Google Scholar 

  46. S. E. Rodabaugh, A categorical accomodation of various notions of fuzzy topology, Fuzzy Sets and Systems 9 (1983), 241–265.

    Article  MathSciNet  MATH  Google Scholar 

  47. S. E. Rodabaugh, A theory of fuzzy uniformities with applications to the fuzzy real lines, J. Math. Anal. Appl. 129 (1988), 37–70.

    Article  MathSciNet  MATH  Google Scholar 

  48. S. E. Rodabaugh, Categorical frameworks for Stone representation theories, Chapter 7 in: S. E. Rodabaugh, E. P. Klement, U. Höhle, eds, Applications Of Category Theory To Fuzzy Subsets, Kluwer Academic Publishers (Boston/Dordrecht/London), 1992, pp. 177–232.

    Google Scholar 

  49. S. E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology,Chapter 4 in: U. Höhle, S. E. Rodabaugh, eds, Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Volume 3(1999), Kluwer Academic Publishers (Dordrecht/Boston/London), pp. 273–388.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  51. V. Šalij, Binary G-relations, Izv. Vysh. Ucheb. Zaved. (Ser. Math.) 1: 44 (1965), 133–145 ( in Russian).

    Google Scholar 

  52. B. Šchweitzer, A Sklar, Probabilistic Metric Spaces, North Holland (Amsterdam), 1983.

    Google Scholar 

  53. A. Šostak, On a fuzzy topological structure, Suppl. Rend. Circ. Matem. Palermo (Ser. II) 11 (1985), 89–103.

    MATH  Google Scholar 

  54. A. Šostak, Coreflectivity in categories of fuzzy topology, in: Continuous Functions on Topological Spaces, Latv. Univ. Publ. ( Riga ) (1986), 159–165 (in Russian).

    Google Scholar 

  55. A. Šostak, On a category for fuzzy topology, Zb. Radova Univ. u Nisu (Ser. Matem.) 2 (1988), 61–67.

    Google Scholar 

  56. A. Šostak, Two decades of fuzzy topology: basic ideas, notions, and results, Russian Math. Surveys 44: 6 (1989), 125–186.

    Article  MathSciNet  MATH  Google Scholar 

  57. A. Šostak, Towards the concept of a fuzzy category, Acta Univ. Latviensis (Ser. Math.) 562 (1991), 85–94.

    Google Scholar 

  58. A. Šostak, On a concept of a fuzzy category, in: 14th Linz Seminar on Fuzzy Set Theory: Non-Classical Logics and Their Applications, Linz, Austria, 1992, 63–66.

    Google Scholar 

  59. A. Šostak, Basic structures of fuzzy topology, J. Math. Sci. textbf 78: 6 (1996), 662–701.

    Article  MATH  Google Scholar 

  60. A. Šostak, On a fuzzy syntopogeneous structure, Quaestiones Mathematicae 20: 3 (1997), 431–463.

    Article  MathSciNet  MATH  Google Scholar 

  61. A. Šostak, Fuzzy categories versus categories of fuzzily structured sets: Elements of the theory of fuzzy categories, Mathematik-Arbeitspapiere 48: Categorical Methods In Algebra And Topology (A collection of papers in honor of Horst Herrlich), Hans-E. Porst ed., pp. 407–438, Bremen, August 1997.

    Google Scholar 

  62. A. Šostak, Fuzzy categories versus categories of fuzzily structured sets: Elements of the theory of fuzzy categories, Part II,Acta Univ. Latv. (Riga, Latvia) (2001), to appear.

    Google Scholar 

  63. A. Šostak, Fuzzy categories related to algebra and topology, Tatra Mount. Math. Publ. 16 (1999), 159–186.

    Google Scholar 

  64. R. H. Warren, Neighbourhoods, bases and continuity in fuzzy topological spaces, Rocky Mount. J. Math. 8 (1978), 459–470.

    MATH  Google Scholar 

  65. Wang Guojun, Topological molecular lattices,Fuzzy Sets and Systems 47 (1992), 351–376.

    Google Scholar 

  66. Yuan Bo, Wu Wangming, Fuzzy ideals on a distributive lattice, Fuzzy Sets and Systems 35 (1990), 231–240.

    Article  MathSciNet  MATH  Google Scholar 

  67. Ying Mingsheng, A new approach to fuzzy topology, Part I, Fuzzy Sets and Systems 39 (1991), 303–321.

    Article  MathSciNet  MATH  Google Scholar 

  68. Ying Mingsheng, A new approach to fuzzy topology, Part II, Fuzzy Sets and Systems 47 (1992), 221–232.

    Article  MathSciNet  MATH  Google Scholar 

  69. Ying Mingsheng, A new approach to fuzzy topology, Part III, Fuzzy Sets and Systems 55 (1993), 193–207.

    Google Scholar 

  70. Ying Mingsheng, Fuzzifying uniform spaces, Fuzzy Sets and Systems 53 (1993), 93–104.

    Google Scholar 

  71. Ying Mingsheng, Fuzzifying topology based on a residuated lattice-valued logic I, Fuzzy Sets and Systems 56 (1993), 337–373.

    Google Scholar 

  72. Ying Mingsheng, Fuzzy topology based on a residuated lattice-valued logic,preprint.

    Google Scholar 

  73. L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338–365.

    Article  MATH  Google Scholar 

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  1. A. P. Šostak
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Editors and Affiliations

  1. Department of Mathematics and Statistics, Youngstown State University, 44555-3609, Youngstown, Ohio, USA

    Stephen Ernest Rodabaugh

  2. Institut für Algebra, Stochastik und wissensbasierte mathematische Systeme, Johannes Kepler Universität, 4040, Linz, Austria

    Erich Peter Klement

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Šostak, A.P. (2003). On Some Fuzzy Categories Related To Category L-TOP Of L-Topological Spaces. In: Rodabaugh, S.E., Klement, E.P. (eds) Topological and Algebraic Structures in Fuzzy Sets. Trends in Logic, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0231-7_14

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  • DOI: https://doi.org/10.1007/978-94-017-0231-7_14

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