Rad HAZU, Matematičke znanosti, Vol. 21 (2017), 143-168.

BORNOLOGICAL STRUCTURES ON MANY-VALUED SETS

Alexander Šostak and Ingrīda Uļjane

Department of Mathematics, University of Latvia, LV-1002 Riga, Latvia
and Institute of Mathematics and Computer Sciences UL, LV-1459 Riga, Latvia
e-mail: aleksandrs.sostaks@lu.lv, aleksandrs.sostaks@lumii.lv

Department of Mathematics, University of Latvia, LV-1002 Riga, Latvia
and Institute of Mathematics and Computer Sciences UL, LV-1459 Riga, Latvia
e-mail: ingrida.uljane@lu.lv, ingrida.uljane@lumii.lv


Abstract.   We introduce an approach to the concept of bornology in the framework of many-valued mathematical structures and develop the basics of the theory of many-valued bornological spaces and initiate the study of the category of many-valued bornological spaces and appropriately defined bounded "mappings" of such spaces. A scheme for constructing many-valued bornologies with prescribed properties is worked out. In particular, this scheme allows to extend an ordinary bornology of a metric space to a many-valued bornology on it.

2010 Mathematics Subject Classification.   54A40.

Key words and phrases.   Bornology, quantale, many-valued mathematical structures, many-valued bornology, fuzzy set.


Full text (PDF) (free access)

DOI: http://doi.org/10.21857/90836cdw6y


References:

  1. M. Abel and A. Šostak, Towards the theory of L-bornological spaces, Iran. J. Fuzzy Syst. 8 (2011) 19-28.
    MathSciNet

  2. G. Beer, S. Naimpally and J. Rodríguez-López, S-topologies and bounded convergences, J. Math. Anal. Appl. 339 (2008), 542-552.
    MathSciNet     CrossRef

  3. G. Beer and S. Levi, Gap, excess and bornological convergence, Set-Valued Anal. 16 (2008), 489-506.
    MathSciNet     CrossRef

  4. G. Beer and S. Levi, Strong uniform continuity, J. Math. Anal. Appl. 350 (2009) 568-589.
    MathSciNet     CrossRef

  5. G. Beer and S. Levi, Total boundedness and bornology, Topology Appl. 156 (2009), 1271-1288.
    MathSciNet     CrossRef

  6. U. Bodenhofer, Ordering of fuzzy sets based on fuzzy orderings. I: The basic approach. Mathware Soft Comput. 15 (2008) 201-218.
    MathSciNet

  7. C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182-190.
    MathSciNet     CrossRef

  8. M. Demirci, Fuzzy functions and their fundamental properties, Fuzzy Sets and Systems 106 (1999), 239-246.
    MathSciNet     CrossRef

  9. M. Demirci, Fundamentals of M-vague algebra and M-vague arithmetic operations, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 10 (2002), 25-75.
    MathSciNet     CrossRef

  10. R. Engelking, General Topology, Panstwowe Wydawnictwo Naukowe, Warszawa, 1977.
    MathSciNet

  11. G. Gierz, K. H. Hoffman, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003.
    MathSciNet     CrossRef

  12. J. A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967), 145-174.
    MathSciNet     CrossRef

  13. J. A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl. 43 (1973), 734-742.
    MathSciNet     CrossRef

  14. H. Hogbe-Nlend, Bornology and Functional Analysis, Math. Studies 26, North-Holland, Amsterdam, 1977.
    MathSciNet &

  15. U. Höhle, Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl. 78 (1980), 659-673.
    MathSciNet     CrossRef

  16. U. Höhle, M-valued sets and sheaves over integral commutative cl-monoids, Chapter 2 in: Applications of Category Theory to Fuzzy Sets, S. E. Rodabaugh, E. P. Klement and U. Höhle (Eds.), Kluwer Acad. Publ., Dordrecht, 1992, pp. 33-72.
    MathSciNet

  17. U. Höhle, Commutative, residuated l-monoids, in: S. E. Rodabaugh, E. P. Klement and U. Höhle (Eds.), Non-classical logics and their applications to Fuzzy Sets, Kluwer Acad. Publ., Dordrecht, Boston, 1995, pp. 53-106.
    MathSciNet

  18. U. Höhle, Many-valued equalities, singletons and fuzzy partitions, Soft computing 2 (1998), 134-140.
    CrossRef

  19. U. Höhle, Many Valued Topology and its Application, Kluwer Acad. Publ., Boston, Dordrecht, London, 2001.
    MathSciNet     CrossRef

  20. U. Höhle, H.-E. Porst and A. Šostak, Fuzzy functions: a fuzzy extension of the category SET and some related categories, Appl. Gen. Topol. 1 (2000), 115-127.
    MathSciNet     CrossRef

  21. S.-T. Hu, Boundedness in a topological space, J. Math. Pures Appl. 28 (1949), 287-320.
    MathSciNet

  22. S.-T. Hu, Introduction to General Topology, Holden-Day, San Francisco, 1966.
    MathSciNet

  23. F. Klawonn, Fuzzy points, fuzzy relations and fuzzy functions, in: V. Novák, I. Perfilieva (Eds.), Discovering the World with Fuzzy Logic, Physica, Heidelberg, 2000, pp. 431-453.
    MathSciNet

  24. E. P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Acad. Publ., Dordrecht, 2000.
    MathSciNet     CrossRef

  25. T. Kubiak, On Fuzzy Topologies, PhD Thesis, Adam Mickiewicz University, Poznań, Poland, 1985.

  26. A. Lechicki, S. Levi and A. Spakowski, Bornological convergence, J. Math. Anal. Appl. 297 (2004), 751-770.
    MathSciNet     CrossRef

  27. Y.-M. Liu and M.-K. Luo, Fuzzy Topology, Advances in Fuzzy Systems - Applications and Topology, World Scientific, Singapore, New Jersey, London, Hong Kong, 1997.
    MathSciNet

  28. W. Morgan, and R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 (1939) 335-354. Reprinted in: K.Bogart, R. Freese and J. Kung (Eds.), The Dilworth Theorems: Selected Papers of R. P. Dilworth, Birkhäser, Basel, 1990.
    MathSciNet     CrossRef

  29. S. Osçağ, Bornologies and bitopological function spaces, Filomat 27 (2013), 1345-1349.
    MathSciNet     CrossRef

  30. J. Paseka, S. A. Solovyov and M. Stehlik, Lattice-valued bornological systems, Fuzzy Sets and Systems 259 (2015), 68-88.
    MathSciNet     CrossRef

  31. I. Perfilieva, Fuzzy function: Theoretical and practical point of view, in: Proc. 7th Conf. European Society for Fuzzy Logic and Technology, EUSFLAT 2011, Aix-Les-Bains, France, July 18-22, 2011, Atlantis Press, 2011, pp. 480-486.
    MathSciNet     CrossRef

  32. I. Perfilieva and A. Šostak, Fuzzy functions: Basics of the Theory and Applications to Topology, preprint, 2015.

  33. G. N. Raney, A subdirect-union representation for completely distibutive complete lattices, Proc. Amer. Math. Soc. 4 (1953), 518-522.
    MathSciNet     CrossRef

  34. S. E. Rodabaugh, Powerset operator based foundation for point-set lattice-theoretic (POSLAT) fuzzy set theories and topologies, Quaestiones Math. 20 (1997), 463-530.
    MathSciNet

  35. K. I. Rosenthal, Quantales and Their Applications, Pirman Research Notes in Mathematics 234, Longman Scientific & Technical, 1990.
    MathSciNet

  36. H. H. Schaefer, Topological Vector Spaces, Springer-Verlag, 1970.
    MathSciNet

  37. A. Šostak, On a fuzzy topological structure, Suppl. Rend. Circ. Mat. Palermo Ser II 11 (1985), 89-103.
    MathSciNet

  38. A. Šostak, Two decades of fuzzy topology: Basic ideas, notions and results, Russian Math. Surveys 44 (1989), 125-186.
    MathSciNet     CrossRef

  39. A. Šostak, Basic structures of fuzzy topology, J. Math. Sci. 78 (1996), 662-701.
    MathSciNet     CrossRef

  40. A. Šostak, Fuzzy functions and an extension of the category L-TOP of Chang-Goguen L-topological spaces, in: Proceedings of the 9th Prague Topological Symposium (2001), Topol. Atlas, North Bay, ON, 2002, pp. 271-294.
    MathSciNet

  41. A. Šostak and I. Uļjane, Bornological structures in the context of L-fuzzy sets, in: 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013), Atlantis Premium Proceedings, pp. 481-488.

  42. A. Šostak and I. Uļjane, L - valued bornologies on powersets, Fuzzy Sets and Systems 294 (2016), 93-104.
    MathSciNet     CrossRef

  43. A. Šostak and I. Uļjane, LM - fuzzy bornologies on many valued sets, in: Proceedings of V Congress Turkic World Mathematicians (Kyrgyzstan, Bulan-Sogottu, 5-7 June, 2014), A. Borubajev (Ed.), Bishkek, Kyrgyz Mathematical Society, 2014, pp. 32-40.

  44. L. Valverde, On the structure of F-indistinguishibility operators, Fuzzy Sets and Systems 17 (1985), 313-328.
    MathSciNet     CrossRef

  45. M. Ying, A new approach to fuzzy topology I, Fuzzy Sets and Systems 39 (1991), 303-321.
    MathSciNet     CrossRef

  46. L. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353.
    MathSciNet

  47. L. Zadeh Similarity relations and fuzzy orderings, Information Sci. 3 (1971), 177-200.
    MathSciNet

  48. L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning I, II, III, Information Sci. 8-9 (1975), 199-257, 301-357, 43-80.
    MathSciNet


Rad HAZU Home Page