Traditionally, rough sets build upon relations based on ordinary sets, i.e. relations on X as subsets of X×X. A starting point of this paper is the equivalent view on relations as mappings from X to the (ordinary) power set PX. Categorically, P is a set functor, and even more so, it can in fact be extended to a monad (P,η,μ). This is still not enough and we need to consider the partial order (PX,≤). Given this partial order, the ordinary power set monad can be extended to a partially ordered monad. The partially ordered ordinary power set monad turns out to contain sufficient structure in order to provide rough set operations. However, the motivation of this paper goes far beyond ordinary relations as we show how more general power sets, i.e. partially ordered monads built upon a wide range of set functors, can be used to provide what we call rough monads.


Partial Order Lower Approximation Natural Transformation Covariant Functor Topological Approach 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Eklund, P., Galán, M.A., Gähler, W., Medina, J., Ojeda Aciego, M., Valverde, A.: A note on partially ordered generalized terms. In: Proc. of Fourth Conference of the European Society for Fuzzy Logic and Technology and Rencontres Francophones sur la Logique Floue et ses applications (Joint EUSFLAT-LFA 2005), pp. 793–796 (2005)Google Scholar
  2. 2.
    Eklund, P., Gähler, W.: Partially ordered monads and powerset Kleene algebras. In: Proc. 10th Information Processing and Management of Uncertainty in Knowledge Based Systems Conference (IPMU 2004) (2004)Google Scholar
  3. 3.
    Gähler, W.: General Topology – The monadic case, examples, applications. Acta Math. Hungar. 88, 279–290 (2000)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gähler, W., Eklund, P.: Extension structures and compactifications. In: Categorical Methods in Algebra and Topology (CatMAT 2000), pp. 181–205 (2000)Google Scholar
  5. 5.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)MATHGoogle Scholar
  6. 6.
    Järvinen, J.: On the structure of rough approximations. Fundamenta Informaticae 53, 135–153 (2002)MATHMathSciNetGoogle Scholar
  7. 7.
    Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, pp. 3–41. Princeton University Press, Princeton (1956)Google Scholar
  8. 8.
    Kortelainen, J.: A Topological Approach to Fuzzy Sets, Ph.D. Dissertation, Lappeenranta University of Technology, Acta Universitatis Lappeenrantaensis 90 (1999)Google Scholar
  9. 9.
    Manes, E.G.: Algebraic Theories. Springer, Heidelberg (1976)MATHGoogle Scholar
  10. 10.
    Pawlak, Z.: Rough sets. Int. J. Computer and Information Sciences 5, 341–356 (1982)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, London (1982)MATHGoogle Scholar
  12. 12.
    Tarski, A.: On the calculus of relations. J. Symbolic Logic 6, 65–106 (1941)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Patrik Eklund
    • 1
  • M. A. Galán
    • 2
  1. 1.Department of Computing ScienceUmeå UniversitySweden
  2. 2.Department of Applied MathematicsUniversity of MálagaSpain

Personalised recommendations