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Rough Objects in Monoidal Closed Categories

  • Patrik EklundEmail author
  • María-Ángeles Galán-García
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

This chapter will build upon previous achievements on monadic rough objects over the category Set, and show how rough object approximation and algebraic manipulation in general can be enriched by extending constructions to work similarly over monoidal closed categories embracing both algebraic as well as order structures. The chapter will also show how the rough information model in this monoidal closed category extension connects with other information models being relational in their basic original structures. Additionally, the chapter will discuss the potential of real world applications.

Notes

Acknowledgement

Research reported by the second author of this chapter was partially supported by the Spanish project:TIN2015-70266-C2-1-P.

References

  1. 1.
    Banach, S.: Théorie des opérations linéares. Zu Subwencji Funduszu Kultury Narodowej, Warsaw (1932)Google Scholar
  2. 2.
    Bénabou, J.: Catégories avec multiplication. C. R. Acad. Sci. Paris 256, 1887–1890 (1963)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bénabou, J.: Algébre élémentaire dans les catégories avec multiplication. C. R. Acad. Sci. Paris 258, 771–774 (1964)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Böttiger, Y., Al, E.: SFINX—a drug-drug interaction database designed for clinical decision support systems. Eur. J. Clin. Pharmacol. 65(6), 627–633 (2009). https://doi.org/10.1007/s00228-008-0612-5 CrossRefGoogle Scholar
  5. 5.
    Bourbaki, N.: Séminaire de Géométrie Algébrique du Bois Marie - 1963–64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1. Lecture Notes in Mathematics, vol. 269. Springer, Berlin (1972)Google Scholar
  6. 6.
    Bousquet, J., Al, E.: Building bridges for innovation in ageing: synergies between action groups of the EIP on AHA. J. Nutr. Health Aging 21(1), 92–104 (2016). https://doi.org/10.1007/s12603-016-0803-1 CrossRefGoogle Scholar
  7. 7.
    de Vries, M., Al, E.: Fall-risk-increasing drugs: a systematic review and meta-analysis: I. Cardiovascular drugs. J. Am. Med. Dir. Assoc. 19(4), 371.e1–371.e9 (2018). https://doi.org/10.1016/j.jamda.2017.12.013 CrossRefGoogle Scholar
  8. 8.
    Eilenberg, S., MacLane, S.: General theory of natural equivalences. Trans. Am. Math. Soc. 58(2), 231–294 (1945)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Eklund, P.: The syntax of many-valued relations. In: Information Processing and Management of Uncertainty in Knowledge-Based Systems, pp. 61–68. Springer, Berlin (2016). https://doi.org/10.1007/978-3-319-40581-0_6 Google Scholar
  10. 10.
    Eklund, P., Gähler, W.: Contributions to fuzzy convergence. In: Gähler, W., Herrlich, H., Preuß, G. (eds.) Recent Developments of General Topology and Its Applications, International Conference in Memory of Felix Hausdorff (1868–1942), pp. 118–123. Akademie Verlag, Berlin (1992)Google Scholar
  11. 11.
    Eklund, P., Gähler, W.: Partially ordered monads and powerset Kleene algebras. In: Proceedings of the 10th Information Processing and Management of Uncertainty in Knowledge Based Systems Conference (IPMU), vol. 3, pp. 1865—1869 (2004)Google Scholar
  12. 12.
    Eklund, P., Galán, M.A.: Monads can be rough. In: Rough Sets and Current Trends in Computing, pp. 77–84. Springer, Berlin (2006). https://doi.org/10.1007/11908029_9 CrossRefGoogle Scholar
  13. 13.
    Eklund, P., Galán, M.: The rough powerset monad. J. Mult. Valued Log. Soft Comput. 13, 321–334 (2007)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Eklund, P., Galán, M.Á.: Partially ordered monads and rough sets, chapter. In: Transactions on Rough Sets VIII, pp. 53–74. Springer, Berlin (2008)Google Scholar
  15. 15.
    Eklund, P., Galán, M., Ojeda-Aciego, M., Valverde, A.: Set functors and generalised terms. In: Proc. IPMU 2000, 8th Information Processing and Management of Uncertainty in Knowledge-Based Systems Conference, vol. III, pp. 1595–1599 (2000)Google Scholar
  16. 16.
    Eklund, P., Galán, M.A., Karlsson, J.: Rough Monadic Interpretations of Pharmacologic Information. In: ICCS 2007, pp. 108–113. Springer, London (2007). https://doi.org/10.1007/978-1-84628-992-7_15 CrossRefGoogle Scholar
  17. 17.
    Eklund, P., Galán, M., Gähler, W.: Partially ordered monads for monadic topologies, rough sets and kleene algebras. Electron. Notes Theor. Comput. Sci. 225, 67–81 (2009). https://doi.org/10.1016/j.entcs.2008.12.067 CrossRefGoogle Scholar
  18. 18.
    Eklund, P., Galán, M.A., Karlsson, J.: Categorical innovations for rough sets. In: Studies in Computational Intelligence, pp. 45–69. Springer, Berlin (2009). https://doi.org/10.1007/978-3-540-89921-1_2
  19. 19.
    Eklund, P., Galán, M., Helgesson, R., Kortelainen, J.: Fuzzy terms. Fuzzy Sets Syst. 256, 211–235 (2014). https://doi.org/10.1016/j.fss.2013.02.012 MathSciNetCrossRefGoogle Scholar
  20. 20.
    Eklund, P., Galán, M.A., Kortelainen, J., Ojeda-Aciego, M.: Monadic formal concept analysis. In: International Conference on Rough Sets and Current Trends in Computing. Lecture Notes in Computer Science, pp. 201–210. Springer, Berlin (2014). https://doi.org/10.1007/978-3-319-08644-6_21 Google Scholar
  21. 21.
    Eklund, P., Hohle, U., Kortelainen, J.: Modules in health classifications. In: 2017 IEEE International Conference on Fuzzy Systems. IEEE, Piscataway (2017).  https://doi.org/10.1109/fuzz-ieee.2017.8015430
  22. 22.
    Eklund, P., Johansson, M., Kortelainen, J.: The logic of information and processes in system-of-systems applications. In: Soft Computing Applications for Group Decision-Making and Consensus Modeling, pp. 89–102. Springer, Berlin (2017). https://doi.org/10.1007/978-3-319-60207-3_6 Google Scholar
  23. 23.
    Eklund, P., García, J.G., Höhle, U., Kortelainen, J.: Semigroups in complete lattices: quantales, modules and related topics. In: Developments in Mathematics, vol. 58. Springer, Berlin (2018)Google Scholar
  24. 24.
    Gähler, W.: Monadic topology – a new concept of generalized topology. In: Gähler, W., Herrlich, H., Preuß, G. (eds.) Recent Developments of General Topology and Its Applications, pp. 136–149. Akademie Verlag, Berlin (1992)zbMATHGoogle Scholar
  25. 25.
    Goguen, J.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Höhle, U., Stout, L.: Foundations of fuzzy sets. Fuzzy Sets Syst. 40, 257–296 (1991)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kelly, G.M.: Basic Concepts of Enriched Category Theory. Lecture Notes Series, vol. 64. Cambridge University Press, London (1982)Google Scholar
  28. 28.
    MacLane, S.: Natural associativity and commutativity. Rice Univ. Stud. 49, 28–46 (1963)MathSciNetzbMATHGoogle Scholar
  29. 29.
    MacLane, S.: Categorical algebra. Bull. Am. Math. Soc. 71(1), 40–106 (1965)MathSciNetCrossRefGoogle Scholar
  30. 30.
    MacLane, S.: Categories for the Working Mathematician. Springer, Berlin (1971)zbMATHGoogle Scholar
  31. 31.
    Nogues, M., Al, E.: Active and healthy ageing: from health prevention to personal care. CARSAT-MACVIA 7 meeting. Montpellier, 7–8th December 2015. Eur. Geriatr. Med. 8(5–6), 511–519 (2017). https://doi.org/10.1016/j.eurger.2017.07.004 CrossRefGoogle Scholar
  32. 32.
    Seppala, L.J., Al, E.: Fall-risk-increasing drugs: a systematic review and meta-analysis: II. Psychotropics. J. Am. Med. Dir. Assoc. 19(4), 371.e11–371.e17 (2018). https://doi.org/10.1016/j.jamda.2017.12.098 Google Scholar
  33. 33.
    Seppala, L.J., et al.: Fall-risk-increasing drugs: a systematic review and meta-analysis: III. Others. J. Am. Med. Dir. Assoc. 19(4), 372.e1–372.e8 (2018). https://doi.org/10.1016/j.jamda.2017.12.099 Google Scholar
  34. 34.
    Stout, L.N.: The logic of unbalanced subobjects in a category with two closed structures. In: Applications of Category Theory to Fuzzy Subsets, pp. 73–105. Springer, Netherlands (1992). https://doi.org/10.1007/978-94-011-2616-8_4 CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Computing ScienceUmeå UniversityUmeåSweden
  2. 2.Department of Applied MathematicsUniversity of MálagaMálagaSpain

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