Advertisement

Fuzzy Sets pp 13-24 | Cite as

Minkowski Functionals of L-Fuzzy Sets

  • Ulrich Höhle

Abstract

The Minkowski functional or gauge of an ordinary set plays an important part in several areas of mathematics, e.g. in the theory of locally convex spaces, optimization, etc. The aim of this paper is to introduce the concept of Minkowski functionals also in the case of L-fuzzy sets. We establish the theorem that there exists a bisection from the set of all absolutely convex, closed, L-fuzzy O-neighborhoods onto the set of all fuzzy continuous, L-probabilistic semi-norms (see Section B).

Keywords

Aequationes Math Complete Boolean Algebra MINKOWSKI Functional Fuzzy Topology Fuzzy Topological Space 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. Birkhoff, “Lattice Theory,” Amer. Math. Soc. Colloquims Publications, Vol. XXV, 3rd Edition, New York (1973).Google Scholar
  2. 2.
    G. Boscan, “Sur certaines semi-normes aléatoires et leurs applications,” C. R. Acad. Sci., Paris 282 (1976), 1319–1321.Google Scholar
  3. 3.
    N. Bourbaki, “Eléments de mathématique, Topologie générale,” Hermann, Paris (1971).Google Scholar
  4. 4.
    C. C. Chang and A. Horn, “On the Representation of α-complete Lattices,” Fundamenta Math. 51 (1962), 253–258.Google Scholar
  5. 5.
    L. Fuchs, “Teilweise geordnete algebraische Strukturen,” Vandenhoeck and Ruprecht, Göttingen (1966).Google Scholar
  6. 6.
    J. A. Goguen, “L-Fuzzy Sets,” J. Math. Anal. Appl. 18 (1967), 145–174.CrossRefGoogle Scholar
  7. 7.
    G. Grätzer, “General Lattice Theory,” Birkhäuser, Basel (1978).Google Scholar
  8. 8.
    U. Höhle, “Probabilistische Topologien,” Manuscripta Math. (1978), 223–245.Google Scholar
  9. 9.
    U. Höhle, “Probabilistische Metriken auf der Menge der nicht negativen Verteilungsfunktionen,” Aequationes Math. 18 (1978), 345–356.CrossRefGoogle Scholar
  10. 10.
    U. Höhle, “Probabilistic Uniformization of Fuzzy Topologies,” Fuzzy Sets and Systems 1 (1978), 311–332.CrossRefGoogle Scholar
  11. 11.
    U. Höhle, “Representation Theorems for L-fuzzy Quantities,” Fuzzy Sets and Systems 3 (to appear).Google Scholar
  12. 12.
    U. Höhle, “Lineare probabilistisch topologische Räume” (unpublished).Google Scholar
  13. 13.
    D. A. Kappos, “Probability Algebras and Stochastic Spaces,” Academic Press, New York, London (1969).Google Scholar
  14. 14.
    A. K. Katsaras and D. B. Liu, “Fuzzy Vector Spaces and Fuzzy Topological Vector Spaces,” J. Math. Anal. Appl. 58 (1977), 135–146.CrossRefGoogle Scholar
  15. 15.
    R. Lowen, “Fuzzy Topological Spaces and Fuzzy Compactness,” J. Math. Anal. Appl. 56 (1976), 621–633.CrossRefGoogle Scholar
  16. 16.
    V. Radu, “Sur une norme aléatoire et la continuité des opérateurs linéaires dans des espaces normes aléatoires,” C. R. Acad. Sci., Paris 280 (1975), 1303–1305.Google Scholar
  17. 17.
    B. Schweizer, “Multiplications on the Space of Probability Distribution Functions,” Aequationes Math. 12 (1975), 156–183.CrossRefGoogle Scholar
  18. 18.
    B. Schweizer and A. Sklar, “Associative Functions and Abstract Semigroups,” Publ. Math. Debrecen 10 (1963), 69–81.Google Scholar
  19. 19.
    R. Sikorski, “Boolean Algebras,” Ergebnisse der Mathematik und ihrer Grenzgebiete, neue Folge Bd. 25, 2nd Edition, Springer, Berlin (1964).Google Scholar
  20. 20.
    L. A. Zadeh, “Fuzzy Sets,” Information and Control 8 (1965), 338–353.CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • Ulrich Höhle
    • 1
  1. 1.Fachbereich MathematikGesamthochschule WuppertalWuppertal 1Federal Republic of Germany

Personalised recommendations