Rectangles, Fringes, and Inverses

  • Gunther Schmidt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4988)


Relational composition is an associative operation; therefore semigroup considerations often help in relational algebra. We study here some less known such effects and relate them with maximal rectangles inside a relation, i.e., with the basis of concept lattice considerations. The set of points contained in precisely one maximal rectangle makes up the fringe. We show that the converse of the fringe sometimes acts as a generalized inverse of a relation. Regular relations have a generalized inverse. They may be characterized by an algebraic condition.


Generalize Inverse Concept Lattice Relation Algebra Interval Order Algebraic Condition 
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.
    Doignon, J.-P., Falmagne, J.-C.: Matching Relations and the Dimensional Structure of Social Sciences. Math. Soc. Sciences 7, 211–229 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ducamp, A., Falmagne, J.-C.: Composite Measurement. J. Math. Psychology 6, 359–390 (1969)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Haralick, R.M.: The diclique representation and decomposition of binary relations. J. ACM 21, 356–366 (1974)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Kim, K.H.: Boolean Matrix Theory and Applications. Monographs and Textbooks in Pure and Applied Mathematics, vol. 70. Marcel Dekker, New York – Basel (1982)Google Scholar
  5. 5.
    Monjardet, B.: Axiomatiques et proprietés des quasi-ordres. Mathematiques et Sciences Humaines 16(63), 51–82 (1978)MathSciNetGoogle Scholar
  6. 6.
    Pirlot, M.: Synthetic description of a semiorder. Discrete Appl. Mathematics 31, 299–308 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Pirlot, M., Vincke, P.: Semiorders — Properties, Representations, Applications. Theory and Decision Library, Mathematical and Statistical Methods, Series B, vol. 36. Kluwer Academic Publishers, Dordrecht (1997)zbMATHGoogle Scholar
  8. 8.
    Schmidt, G., Ströhlein, T.: Relationen und Graphen. Mathematik für Informatiker. Springer, Heidelberg (1989)zbMATHGoogle Scholar
  9. 9.
    Schmidt, G., Ströhlein, T.: Relations and Graphs — Discrete Mathematics for Computer Scientists. In: EATCS Monographs on Theoretical Computer Science. Springer, Heidelberg (1993)Google Scholar
  10. 10.
    Winter, M.: Decomposing Relations Into Orderings. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS/AKA 2003. LNCS, vol. 3051, pp. 261–272. Springer, Heidelberg (2004) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Gunther Schmidt
    • 1
  1. 1.Institute for Software Technology, Department of Computing ScienceUniversität der Bundeswehr MünchenNeubibergGermany

Personalised recommendations