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Variable-basis topological systems versus variable-basis topological spaces

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Abstract

The paper introduces a variable-basis generalization of the notion of topological system of Vickers and considers functorial relationships between the categories of variable-basis topological systems and variable-basis fuzzy topological spaces in the sense of Rodabaugh.

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References

  • Abramsky S, Vickers S (1993) Quantales, observational logic and process semantics. Math Struct Comput Sci 3:161–227

    Article  MATH  MathSciNet  Google Scholar 

  • Adámek J, Rosický J (1994) Locally presentable and accessible categories. Cambridge University Press, Cambridge

  • Adámek J, Herrlich H, Strecker GE (2006) Abstract and concrete categories: the joy of cats. Repr Theory Appl Categ 17:1–507

    Google Scholar 

  • Chang CL (1968) Fuzzy topological spaces. J Math Anal Appl 24:182–190

    Article  MATH  MathSciNet  Google Scholar 

  • Cohn PM (1981) Universal algebra. Mathematics and its applications, vol 6. D. Reidel Publishing Company, Dordrecht

    Google Scholar 

  • Denniston JT, Rodabaugh SE (2008) Functorial relationships between lattice-valued topology and topological systems. Quaest. Math. (to appear)

  • Denniston JT, Melton A, Rodabaugh SE (2009) Lattice-valued topological systems. In: Bodenhofer U, De Baets B, Klement EP, Saminger-Platz S (eds) Abstracts of the 30th Linz seminar on fuzzy set theory. Johannes Kepler Universität, Linz, pp 24–31

  • Eklund P (1984) Category theoretic properties of fuzzy topological spaces. Fuzzy Sets Syst 13:303–310

    Article  MATH  MathSciNet  Google Scholar 

  • Eklund P (1986a) A comparison of lattice-theoretic approaches to fuzzy topology. Fuzzy Sets Syst 19:81–87

    MATH  MathSciNet  Google Scholar 

  • Eklund P (1986b) Categorical fuzzy topology. PhD thesis, Åbo Akademi

  • Goguen JA (1967) L-fuzzy sets. J Math Anal Appl 18:145–174

    Article  MATH  MathSciNet  Google Scholar 

  • Goguen JA (1973) The fuzzy Tychonoff theorem. J Math Anal Appl 43:734–742

    Article  MATH  MathSciNet  Google Scholar 

  • Guido C (2003) Powerset operators based approach to fuzzy topologies on fuzzy sets. In: Rodabaugh SE, Klement EP (eds) Topological and algebraic structures in fuzzy sets. A handbook of recent developments in the mathematics of fuzzy sets. Kluwer Academic Publishers, Dordrecht. Trends Log. Stud. Log. Libr. 20, pp 401–413

  • Guido C (2009) Attachment between fuzzy points and fuzzy sets. In: Bodenhofer U, De Baets B, Klement EP, Saminger-Platz S (eds) Abstracts of the 30th Linz seminar on fuzzy set theory. Johannes Kepler Universität, Linz, pp 52–54

  • Herrlich H, Strecker GE (2007) Category theory, Sig. Ser. Pure Math., vol 1, 3rd edn. Heldermann Verlag, Berlin

  • Höhle U (2007a) Fuzzy sets and sheaves. Part I. Basic concepts. Fuzzy Sets Syst 158(11):1143–1174

    Article  MATH  Google Scholar 

  • Höhle U (2007b) Fuzzy sets and sheaves. Part II. Sheaf-theoretic foundations of fuzzy set theory with applications to algebra and topology. Fuzzy Sets Syst 158(11):1175–1212

    Article  MATH  Google Scholar 

  • Höhle U, Šostak AP (1999) Axiomatic foundations of fixed-basis fuzzy topology. In: Höhle U, Rodabaugh SE (eds) Mathematics of fuzzy sets: logic, topology and measure theory, The handbooks of fuzzy sets series, vol 3. Kluwer Academic Publishers, Dordrecht, pp 123–272

  • Hutton B (1980) Products of fuzzy topological spaces. Topology Appl 11:59–67

    Article  MATH  MathSciNet  Google Scholar 

  • Isbell JR (1972) Atomless parts of spaces. Math Scand 31:5–32

    MATH  MathSciNet  Google Scholar 

  • Johnstone PT (1982) Stone spaces. Cambridge University Press, Cambridge

  • Kotzé W, Kubiak T (1992) Fuzzy topologies of Scott continuous functions and their relation to the hypergraph functor. Quaest Math 15(2):175–187

    MATH  Google Scholar 

  • Kubiak T (1985) On fuzzy topologies. PhD thesis, Adam Mickiewicz Univ., Poznań, Poland

  • Kubiak T, Šostak A (2009) Foundations of the theory of (L,M)-fuzzy topological spaces. In: Bodenhoer U, De Baets B, Klement EP, Saminger-Platz S (eds) Abstracts of the 30th Linz seminar on fuzzy set theory. Johannes Kepler Universität, Linz, pp 70–73

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

    Article  MATH  MathSciNet  Google Scholar 

  • Mac Lane S (1998) Categories for the Working Mathematician, 2nd edn. Springer, Berlin

  • Manes EG (1976) Algebraic theories. Springer, Berlin

  • Papert D, Papert S (1959) Sur les treillis des ouverts et les paratopologies. Semin. de Topologie et de Geometrie differentielle Ch. Ehresmann 1 (1957/58), No. 1, p 1–9

  • Pultr A, Rodabaugh SE (2008a) Category theoretic aspects of chain-valued frames. Part I. Categorical foundations. Fuzzy Sets Syst 159:501–528

    Article  MATH  MathSciNet  Google Scholar 

  • Pultr A, Rodabaugh SE (2008b) Category theoretic aspects of chain-valued frames. Part II. Applications to lattice-valued topology. Fuzzy Sets Syst 159:529–558

    Article  MATH  MathSciNet  Google Scholar 

  • Resende P (2000) Quantales and observational semantics. In: Coecke B, Moore D, Wilce A (eds) Current research in operational quantum logic: algebras, categories and languages. Kluwer Academic Publishers, Dordrecht. Fundam. Theor. Phys., vol 111, pp 263–288

  • Resende P (2001) Quantales, finite observations and strong bisimulation. Theor Comput Sci 254(1–2):95–149

    Article  MATH  MathSciNet  Google Scholar 

  • Rodabaugh SE (1983) A categorical accommodation of various notions of fuzzy topology. Fuzzy Sets Syst 9:241–265

    Article  MATH  MathSciNet  Google Scholar 

  • Rodabaugh SE (1991) Point-set lattice-theoretic topology. Fuzzy Sets Syst 40(2):297–345

    Article  MATH  MathSciNet  Google Scholar 

  • Rodabaugh SE (1992) Categorical frameworks for stone representation theories. In: Rodabaugh SE, Klement EP, Höhle U (eds) Applications of category theory to fuzzy subsets, theory and decision library: Series B: Mathematical and statistical methods, vol 14. Kluwer Academic Publishers, Dordrecht, pp 177–231

  • Rodabaugh SE (1997) Powerset operator based foundation for point-set lattice-theoretic (poslat) fuzzy set theories and topologies. Quaest Math 20(3):463–530

    MATH  MathSciNet  Google Scholar 

  • Rodabaugh SE (1999) Categorical foundations of variable-basis fuzzy topology. In: Höhle U, Rodabaugh SE (eds) Mathematics of fuzzy sets: logic, topology and measure theory, The handbooks of fuzzy sets series, vol 3. Kluwer Academic Publishers, Dordrecht, pp 273–388

  • Rodabaugh SE (2007) Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics. Int J Math Math Sci 2007:Article ID 43,645, 71 pp. doi:10.1155/2007/43645

  • Solovjovs S (2008a) From quantale algebroids to topological spaces. In: Klement EP, Rodabaugh SE, Stout LN (eds) Abstracts of the 29th Linz seminar on fuzzy set theory. Johannes Kepler Universität, Linz, pp 98–101

  • Solovjovs S (2008b) On a categorical generalization of the concept of fuzzy set: basic definitions, properties, examples. VDM Verlag Dr. Müller

  • Solovyov S (2006) On the category Set(JCPos). Fuzzy Sets Syst 157(3):459–465

    Article  MATH  MathSciNet  Google Scholar 

  • Solovyov S (2008a) Categorical frameworks for variable-basis sobriety and spatiality. Math Stud (Tartu) 4:89–103

    MathSciNet  Google Scholar 

  • Solovyov S (2008b) Sobriety and spatiality in varieties of algebras. Fuzzy Sets Syst 159(19):2567–2585

    Article  MATH  MathSciNet  Google Scholar 

  • Vickers S (1989) Topology via Logic. Cambridge University Press, Cambridge

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–365

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This research was supported by the European Social Fund. The author is also grateful to the Department of Mathematics “Ennio De Giorgi” of the University of Salento in Lecce, Italy (especially to Prof. C. Guido) for the opportunity of spending a month at the university during which period the revised version of the manuscript was prepared.

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  1. Department of Mathematics, University of Latvia, Zellu iela 8, Riga, 1002, Latvia

    Sergey A. Solovyov

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  1. Sergey A. Solovyov
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Solovyov, S.A. Variable-basis topological systems versus variable-basis topological spaces. Soft Comput 14, 1059–1068 (2010). https://doi.org/10.1007/s00500-009-0485-2

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