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Continuous triangular norm based fuzzy topology

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Abstract

For each continuous t-norm &, a class of fuzzy topological spaces, called &-topological spaces, is introduced. The motivation stems from the idea that to each many-valued logic there may correspond a theory of many-valued topology, in particular, each continuous t-norm may lead to a theory of fuzzy topology. It is shown that for each continuous t-norm &, the subcategory consisting of &-topological spaces is simultaneously reflective and coreflective in the category of fuzzy topological spaces, hence gives rise to an autonomous theory of fuzzy topology. Topologizing a fuzzy pre-ordered set with the fuzzy Scott topology yields a functor from the category of fuzzy pre-ordered sets and maps that preserve suprema of flat ideals to the category of &-topological spaces. It is proved that this functor is a full one if and only if the t-norm & is Archimedean.

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References

  1. Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publishers, Dordrecht (2002)

    Book  MATH  Google Scholar 

  2. Bonsangue, M.M., van Breugel, F., Rutten, J.J.M.M.: Generalized metric space: completion, topology, and powerdomains via the Yoneda embedding. Theor. Comput. Sci. 193, 1–51 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, P., Lai, H., Zhang, D.: Coreflective hull of finite strong \(L\)-topological spaces. Fuzzy Sets Syst. 182, 79–92 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cignoli, R., Esteva, F., Godo, L., Torrens, A.: Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Comput. 4, 106–112 (2000)

    Article  Google Scholar 

  5. Cintula, P., Hájek, P.: Triangular norm based predicate fuzzy logics. Fuzzy Sets Syst. 161, 311–346 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Flagg, R.C., Sünderhauf, P.: The essence of ideal completion in quantitative form. Theor. Comput. Sci. 278, 141–158 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Flagg, R.C., Sünderhauf, P., Wagner, K.R.: A logical approach to quantitative domain theory. Topology Atlas Preprint No. 23 (1996). http://at.yorku.ca/e/a/p/p/23.htm

  8. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Encyclopedia of Mathematics and Its Applications, vol. 93. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  9. Goubault-Larrecq, J.: Non-Hausdorff Topology and Domain Theory. Cambridge University Press, Cambridge (2013)

    Book  MATH  Google Scholar 

  10. Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  11. Hájek, P.: What is mathematical fuzzy logic. Fuzzy Sets Syst. 157, 597–603 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Höhle, U.: Probabilistische Topologien. Manuscr. Math. 26, 223–245 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Höhle, U.: Characterization of \(L\)-topologies by \(L\)-valued neighborhoods. In: Höhle, U., Rodabaugh, S. (eds.) Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, pp. 389–432. Springer, Berlin (1999)

    Chapter  Google Scholar 

  14. Höhle, U., S̆ostak, A.P.: Axiomatic foundation of fixed-basis fuzzy topology. In: Höhle, U., Rodabaugh, S.E. (eds.) Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, pp. 123–272. Springer, Berlin (1999)

    Chapter  Google Scholar 

  15. Kelly, G.M.: Basic Concepts of Enriched Category Theory. London Mathematical Society Lecture Notes Series, vol. 64. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  16. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms, Trends in Logic, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  17. Lai, H., Zhang, D.: Fuzzy preorder and fuzzy topology. Fuzzy Sets Syst. 157, 1865–1885 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lai, H., Zhang, D.: Complete and directed complete \(\Omega \)-categories. Theor. Comput. Sci. 388, 1–25 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lai, H., Zhang, D.: Closedness of the category of liminf complete fuzzy orders. Fuzzy Sets Syst. 282, 86–98 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lai, H., Zhang, D.: Fuzzy topological spaces with conical neighborhood systems. Fuzzy Sets Syst. 330, 87–104 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lai, H., Zhang, D., Zhang, G.: A comparative study of ideals in fuzzy orders. Fuzzy Sets Syst. (2018). https://doi.org/10.1016/j.fss.2018.11.019

    Google Scholar 

  22. Lai, H., Zhang, D., Zhang, G.: Flat ideals in the unit interval with the canonical fuzzy order. J. Sichuan Univ. (Nat. Sci. Ed.) 55, 905–911 (2018)

    MATH  Google Scholar 

  23. Lawvere, F.W.: Metric spaces, generalized logic, and closed categories. Rendiconti del Seminario Matématico e Fisico di Milano 43, 135–166 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, W., Lai, H., Zhang, D.: Yoneda completeness and flat completeness of ordered fuzzy sets. Fuzzy Sets Syst. 313, 1–24 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lowen, R.: Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 56, 623–633 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lowen, R., Wuyts, P.: Stable subconstructs in FTS I. J. Fuzzy Math. 1, 475–489 (1993)

    MathSciNet  MATH  Google Scholar 

  27. Lowen, R., Wuyts, P.: Stable subconstructs in FTS II. Fuzzy Sets Syst. 76, 169–189 (1995)

    Article  MATH  Google Scholar 

  28. Mostert, P.S., Shields, A.L.: On the structure of semigroups on a compact manifold with boundary. Ann. Math. 65, 117–143 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  29. Novák, V., De Baets, B.: EQ-algebras. Fuzzy Sets Syst. 160, 2956–2978 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Takeuti, G., Titani, S.: Fuzzy logic and fuzzy set theory. Arch. Math. Log. 32, 1–32 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tao, Y., Lai, H., Zhang, D.: Quantale-valued preorders: globalization and cocompleteness. Fuzzy Sets Syst. 256, 236–251 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Vickers, S.: Localic completion of generalized metric spaces. Theory Appl. Categ. 14, 328–356 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Wagner, K.R.: Liminf convergence in \(\Omega \)-categories. Theor. Comput. Sci. 184, 61–104 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yao, W.: A categorical isomorphism between injective fuzzy \(T_0\)-spaces and fuzzy continuous lattices. IEEE Trans. Fuzzy Syst. 24, 131–139 (2016)

    Article  Google Scholar 

  35. Zadeh, L.A.: Similarity relations and fuzzy orderings. Inf. Sci. 3, 177–200 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang, D.: An enriched category approach to many valued topology. Fuzzy Sets Syst. 158, 349–366 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank sincerely Dr. Hongliang Lai for stimulating discussion during the preparation of this paper.

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  1. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Dexue Zhang & Gao Zhang

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  1. Dexue Zhang
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  2. Gao Zhang
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Correspondence to Dexue Zhang.

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This work is supported by National Natural Science Foundation of China (No. 11871358).

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Zhang, D., Zhang, G. Continuous triangular norm based fuzzy topology. Arch. Math. Logic 58, 915–942 (2019). https://doi.org/10.1007/s00153-019-00670-1

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