Fuzzy Real Lines And Dual Real Lines As Poslat Topological, Uniform, And Metric Ordered Semirings With Unity

  • S. E. Rodabaugh
Part of the The Handbooks of Fuzzy Sets Series book series (FSHS, volume 3)


Nontrivial examples of objects and morphisms are fundamentally important to establishing the credibility of a new category or discipline such as lattice-dependent or fuzzy topology; and often the justifications of the importance of certain objects and the importance of certain morphisms are intertwined. In [33], we established classes of variable-basis morphisms between different fuzzy real lines and between different dual real lines, but left untouched the issue of the canonicity of these objects. In this chapter, we attempt to demonstrate the canonicity of these spaces stemming from the interplay between arithmetic operations and underlying topological structures. We shall summarize the definitions of fuzzy addition and fuzzy multiplication on the fuzzy real lines and indicate their joint-continuity—along with that of the addition and multiplication on the usual real line—with respect to the underlying poslat topologies, as well as the quasi-uniform and uniform continuity (in the case of fuzzy addition and addition) with respect to the underlying quasi-uniform, uniform, and metric structures. These results not only help establish fuzzy topology w.r.t. objects, but enrich our understanding of traditional arithmetic operations.


Fuzzy Number Arithmetic Operation Uniform Continuity Crisp Number Fundamental Identity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    D. Dubois and H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci. 9 (1978), 613–626.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    D. Dubois and H. Prade, Fuzzy real algebra: some results, Fuzzy Sets and Systems 2 (1979), 327–348.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    M. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1979), 205–230.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    T. E. Gantner, R. C. Steinlage, and R. H. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978), 547–562.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    R. Goetschel, Jr. and W. Voxman, Topological properties of fuzzy numbers, Fuzzy Sets and Systems 10 (1983), 87–99.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    J. A. Goguen, The fuzzy Tychonoff Theorem, J. Math. Anal. Appl. 43 (1973), 734–742.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    U. Höhle, Probabilitische Metriken auf der Menge der nicht negativen Verteilungsfunktionen, Aequationes Mathematicae 18 (1978), 345–356.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    U. Höhle and A. Šostak, Axiomatic foundations of fixed-basis fuzzy topology (Chapter 3 in this Volume).Google Scholar
  9. [9]
    B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl. 50 (1975), 74–79.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    B. Hutton, Uniformities on fuzzy topological spaces, J. Math. Anal. Appl. 58 (1977), 559–571.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    A. J. Klein, α-Closure in fuzzy topology, Rocky Mount. J. Math. 11 (1981) 553–560.MATHCrossRefGoogle Scholar
  12. [12]
    A. J. Klein, Generating fuzzy topologies with semi-closure operators, Fuzzy Sets and Systems 9 (1983), 267–274.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    A. J. Klein, Generalizing the L-fuzzy unit interval, Fuzzy Sets and Systems 12 (1984), 271–279.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    W. Kotzé, Uniform spaces (Chapter 8 in this Volume).Google Scholar
  15. [15]
    W. Kotzé and T. Kubiak, Fuzzy topologies of Scott continuous functions and their relationship to the hypergraph functor, Quaestiones Mathemati-cae 15(1992), 175–187.MATHCrossRefGoogle Scholar
  16. [16]
    W. Kotzé and T. Kubiak, Inserting L-fuzzy-valued functions, Mathematische Nachrichten 164(1993), 5–11.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    T. Kubiak, Extending continuous L-real valued functions, Math. Japon 31(1986), 875–887.MathSciNetMATHGoogle Scholar
  18. [18]
    T. Kubiak, L-fuzzy normal spaces and Tietze Extension Theorem, J. Math. Anal. Appl. 125 (1987), 141–153.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    T. Kubiak, The fuzzy unit interval and the Helly space, Math. Japonica 33 (1988), 253–259.MathSciNetMATHGoogle Scholar
  20. [20]
    T. Kubiak, The topological modification of the L-fuzzy unit interval, in: “Applications of Category Theory to Fuzzy Subsets”, S. E. Rodabaugh et al., editors, (Kluwer Academic Publishers, Dordrecht 1992).Google Scholar
  21. [21]
    T. Kubiak, On L-Tychonoff spaces, Fuzzy Sets and Systems 73(1995), 25–53.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    T. Kubiak, Separation axioms: extension of mappings and embeddings of spaces (Chapter 6 in this Volume).Google Scholar
  23. [23]
    R. Lowen, A comparison of different compactness notions in fuzzy topological spaces, J. Math. Anal. Appl. 64 (1978), 446–454.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    R. Lowen, On (ℝ(L),⊕), Fuzzy Sets and Systems 10 (1983), 203–209.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    S. E. Rodabaugh, The Hausdorff separation axiom for fuzzy topological spaces, Topology and its Applications 11 (1980), 319–334.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    S. E. Rodabaugh, Fuzzy addition in the L-fuzzy real line, Fuzzy Sets and Systems 8 (1982), 39–51.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    S. E. Rodabaugh, Separation axioms and the L-fuzzy real lines, Fuzzy Sets and Systems 11 (1983), 163–183.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    S. E. Rodabaugh, Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines, Fuzzy Sets and Systems 15 (1985), 285–310.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    S. E. Rodabaugh, A theory of fuzzy uniformities with applications to the fuzzy real lines, J. Math. Anal. Appl. 129 (1988), 37–70.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    S. E. Rodabaugh, Dynamic topologies and their applications to crisp topologies, fuzzification of crisp topologies, and fuzzy topologies on the crisp real line, J. Math. Anal. Appl. 131 (1988), 25–66.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    S. E. Rodabaugh, Lowen, para-Lowen, and α-level functors and fuzzy topologies on the crisp real line, J. Math. Anal. Appl. 131 (1988), 157–169.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    S. E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets and Systems 40 (1991), 297–345.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    S. E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology(Chapter 4 in this Volume).Google Scholar
  34. [34]
    S. E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies (Chapter 2 in this Volume).Google Scholar
  35. [35]
    S. E. Rodabaugh, Separation axioms: representation theorems, compactness, and compactifications (Chapter 7 in this Volume).Google Scholar
  36. [36]
    E. S. Santos, Topology versus fuzzy topology, Preprint, Youngstown State University (1977).Google Scholar
  37. [37]
    C. K. Wong, Fuzzy topology: product and quotient theorems, J. Math. Anal. Appl. 45 (1974), 512–521.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • S. E. Rodabaugh
    • 1
  1. 1.Department of Mathematics and StatisticsYoungstown State UniversityYoungstownUSA

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