Some Remarks on Approximations of Arbitrary Binary Relations by Partial Orders

  • Ryszard Janicki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5306)


When a non-numerical ranking is created using Pairwise Comparisons paradigm, its first estimation is a binary relation which may not even be a partial order. In the paper four different partial order approximations of an arbirary binary relation are introduced and discussed.


Partial Order Transitive Closure Weak Order Inclusion Property Arbitrary Relation 
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.
    Arrow, K.J.: Social Choice and Individual Values. J. Wiley, Chichester (1951)MATHGoogle Scholar
  2. 2.
    Dyer, J.S.: Remarks on the Analytic Hierarchy Process. Management Sci. 36, 244–258 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fishburn, P.C.: Interval Orders and Interval Graphs. J. Wiley, New York (1985)MATHGoogle Scholar
  4. 4.
    French, S.: Decision Theory. Ellis Horwood, New York (1986)MATHGoogle Scholar
  5. 5.
    Janicki, R.: Pairwise Comparisons, Incomparability and Partial Orders. In: Proc. of ICEIS 2007 (Int. Conf. on Enterprise Information Systems), Funchal, Portugal, vol. 2, pp. 297–302 (2007)Google Scholar
  6. 6.
    Janicki, R.: Ranking with Partial Orders and Pairwise Comparisons. In: Wang, G., Li, T., Grzymała-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 442–451. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Janicki, R., Koczkodaj, W.W.: Weak Order Approach to Group Ranking. Computers Math. Applic. 32(2), 51–59 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Pawlak, Z.: Rough Sets. Kluwer, Dordrecht (1991)CrossRefMATHGoogle Scholar
  9. 9.
    Rosen, K.H.: Discrete Mathematics and Its Applications. McGraw-Hill, New York (1999)Google Scholar
  10. 10.
    Saaty, T.L.: A Scaling Methods for Priorities in Hierarchical Structure. Journal of Mathematical Psychology 15, 234–281 (1977)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Schröder, E.: Algebra der Logik, 2nd edn., Teuber, Leipzig, vol. 1895. Chelsea (1966)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ryszard Janicki
    • 1
  1. 1.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada

Personalised recommendations