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Modules in the Category \(\mathtt {\mathbf{Sup}}\)

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On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 336))

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Abstract

This chapter explains basic properties of left modules on unital quantales with the perspective towards fuzzy set theory. Typical constructions such as the fuzzy power set, Zadeh’s forward operator or binary operations defined according to Zadeh’s extension principle are constructions in the symmetric monoidal closed category of complete lattices and join preserving maps. Moreover, involutive left modules play a significant role in the representation theory of \(C^*\)-algebras.

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References

  1. Adámek, J.: Free algebras and automata realization in the language of categories. Comment Math. Univ. Carol. 15, 589–602 (1974)

    MATH  Google Scholar 

  2. Adámek, J., Trnková, V.: Automata and Algebras in Categories. Kluwer Academic Publishers, Dordrecht (1990)

    MATH  Google Scholar 

  3. Banaschewski, B., Nelson, E.: Tensor products and bimorphisms. Cand. Math. Bull. 19, 385–402 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Birkhoff, G.: Lattice Theory, Colloquium Publications, vol. 25, 3rd edn., eighth printing. American Mathematical Society, Rhode Island (1995)

    Google Scholar 

  5. Denniston, J.T., Melton, A., Rodabaugh, S.E.: Enriched categories and many-valued preorders: categorical, semantical and topological perspectives. Fuzzy Sets Syst. 256, 4–56 (2014)

    Article  MathSciNet  Google Scholar 

  6. Eklund, P., Höhle, U., Kortelainen, J.: A survey on the categorical term construction with applications. Fuzzy Sets and Syst. doi:10.1016/j.fss.2015.07.003

    Google Scholar 

  7. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Studies in Logic, vol. 151. Elsevier, Amsterdam (2007)

    Google Scholar 

  8. Gutiérrez García, J., Höhle, U., Kubiak, T.: Tensor products in \(\mathtt{Sup}\) and their application in constructing quantales (submitted)

    Google Scholar 

  9. Höhle, U.: Many Valued Topology and Its Applications. Kluwer Academic Publishers, Boston (2001)

    Book  MATH  Google Scholar 

  10. Höhle, U.: Categorical foundations of topology with applications to quantaloid enriched topological spaces. Fuzzy Sets Syst. 256, 166–210 (2014)

    Article  Google Scholar 

  11. Höhle, U.: Many-valued preorders I: the basis of many-valued mathematics. In: Magdalena, L., et al. (eds.) Enric Trillas: A Passion for Fuzzy Sets, Studies in Fuzziness and Soft Computing, vol. 322, pp. 125–150. Springer, Heidelberg (2015)

    Google Scholar 

  12. Joyal, A., Tierney, M.: An Extension of the Galois Theory of Grothendieck, Memoirs of the American Mathematical Society, vol. 51, Number 309. American Mathematical Society (1984)

    Google Scholar 

  13. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, Volume I Elementary Theory, Graduate Studies in Mathematics Volume 15. American Mathematical Society (1997)

    Google Scholar 

  14. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, Volume II Adavanced Theory, Graduate Studies in Mathematics Volume 16. American Mathematical Society (1997)

    Google Scholar 

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

    Google Scholar 

  16. Kruml, D., Resende, P.: On quantales that classify \(C^*\)-algebras. Cahiers Topol. Géom. Différ. Catég. 45, 287–296 (2004)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  18. Manes, E.G.: Algebraic Theories. Springer, New York (1976)

    Book  MATH  Google Scholar 

  19. Meise, R., Vogt, D.: Introduction to Functional Analysis, Oxford Gruaduate Texts in Mathematics. Oxford University Press (1997)

    Google Scholar 

  20. Mulvey, C.J., Pelletier, J.W.: A quantisation of the calculus of relations, CMS Proceedings, vol. 13, pp. 345–360. American Mathematical Society, Providence (1992)

    Google Scholar 

  21. Mulvey, C.J., Pelletier, J.W.: On the quantisation of points. J. Pure Appl. Algebra 159, 231–295 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pelletier, J.W., Rosický, J.: Simple involutive quantales. J. Algebra 195, 367–386 (1987)

    Article  Google Scholar 

  23. Pu, Q., Zhang, D.: Preordered sets valued in a \(GL\)-monoid. Fuzzy Sets Syst. 187, 1–32 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Resende, P.: Sup-lattice 2-forms and quantales. J. Algebra 276, 143–167 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rodabaugh, S.E.: Powerset operator foundations for poslat fuzzy theories and topologies. In: Höhle, U., Rodabaugh, S.E. (eds.) Logic, Topology, Theory, Measure, Mathematics of Fuzzy Sets, pp. 91–116. Kluwer Academic Publishers (1999)

    Google Scholar 

  26. Rosenthal, K.I.: Quantales and Their Applications, Pitman Research Notes in Mathematics, vol. 234. Longman Scientific Technical, Longman House, Burnt Mill, Harlow (1990)

    Google Scholar 

  27. Shmuely, Z.: The structure of Galois connections. Pac. J. Math. 54, 209–225 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  28. Solovyov, S.A.: Powerset operator foundation for catalg fuzzy set theories. Iran. J. Fuzzy Syst. 8, 1–46 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Stubbe, I.: Categorical structures enriched in a quantaloid tensored and cotensored categories. Theory Appl. Categ. 16, 283–306 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning I. Inf. Sci. 8, 119–249 (1975)

    Google Scholar 

  32. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning II. Inf. Sci. 8, 301–357 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning III. Inf. Sci. 9, 43–80 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Fachbereich C Mathematik und Naturwissenschaften, Bergische Universität, Wuppertal, Germany

    Ulrich Höhle

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  1. Ulrich Höhle
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Correspondence to Ulrich Höhle .

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Editors and Affiliations

  1. Dept of Knowledge-Based Math. Systems, Johannes Kepler University Linz, Linz, Austria

    Susanne Saminger-Platz

  2. Dept. of Mathematics & Descriptive Geo., Slovak University of Technology, BRATISLAVA, Slovakia

    Radko Mesiar

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Höhle, U. (2016). Modules in the Category \(\mathtt {\mathbf{Sup}}\) . In: Saminger-Platz, S., Mesiar, R. (eds) On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 336. Springer, Cham. https://doi.org/10.1007/978-3-319-28808-6_3

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  • DOI: https://doi.org/10.1007/978-3-319-28808-6_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28807-9

  • Online ISBN: 978-3-319-28808-6

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