Eigenproblem in Max-Drast and Max-Łukasiewicz Algebra

Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 677)


When the max-min operations on the unit real interval are considered as a particular case of fuzzy logic operations (Gödel operations), then the max-min algebra can be viewed as a specific case of more general fuzzy algebra with operations max and T, where T is a triangular norm (in short: t-norm). Such max-T algebras are useful in various applications of the fuzzy set theory. In this chapter we investigate the structure of the eigenspace of a given fuzzy matrix in two specific max-T algebras: the so-called max-drast algebra , in which the least t-norm T (often called the drastic norm) is used, and max-Lukasiewicz algebra with Łukasiewicz t-norm L. For both of these max-T algebras the necessary and sufficient conditions are presented under which the monotone eigenspace (the set of all non-decreasing eigenvectors) of a given matrix is non-empty and, in the positive case, the structure of the monotone eigenspace is described. Using permutations of matrix rows and columns, the results are extended to the whole eigenspace.


Fuzzy Transition Matrix Triangular Norms Fuzzy Algebra Eigenspace Structure General Eigenproblem 
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.


  1. 1.
    Casasnovas, J., Mayor, G.: Discrete t-norms and operations on extended multisets. Fuzzy Sets Syst. 159, 1165–1177 (2008)CrossRefGoogle Scholar
  2. 2.
    Cechlárová, K: Eigenvectors in bottleneck algebra. Lin. Algebra Appl. 175, 63–73 (1992)CrossRefGoogle Scholar
  3. 3.
    Cechlárová, K: Efficient computation of the greatest eigenvector in fuzzy algebra. Tatra Mt. Math. Publ. 12, 73–79 (1997)Google Scholar
  4. 4.
    Cohen, G., Dubois, D., Quadrat, J.P., Viot, M.: A linear-system-theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automat. Contr. AC-30, 210–220 (1985)CrossRefGoogle Scholar
  5. 5.
    Cuninghame-Green, R.A.: Describing industrial processes with interference and approximating their steady-state behavior. Oper. Res. Quart. 13, 95–100 (1962)CrossRefGoogle Scholar
  6. 6.
    Cuninghame-Green, R.A.: Minimax Algebra, Lecture Notes in Economics and Mathematical Systems, vol. 166. Springer, Berlin (1979)Google Scholar
  7. 7.
    Cuninghame-Green, R.A.: Minimax algebra and application. In: Hawkes, P.W. (ed.), Advances in Imaging and Electron Physics, vol. 90. Academic Press, New York (1995)Google Scholar
  8. 8.
    Gavalec, M.: Monotone eigenspace structure in max–min algebra. Lin. Algebra Appl. 345, 149–167 (2002)CrossRefGoogle Scholar
  9. 9.
    Gavalec, M., Rashid, I.: Monotone eigenspace structure of a max-drast fuzzy matrix. In: Proc. of the 28th Int. Conf. Mathematical Methods in Economics, University of South Bohemia, České Budějovice 2010, pp. 162–167 (2010)Google Scholar
  10. 10.
    Gavalec, M., Rashid, I., Cimler, R.: Eigenspace structure of a max-drast fuzzy matrix. Fuzzy Sets Syst. (2013). Doi:10.1016/j.fss.2013.10.008Google Scholar
  11. 11.
    Gondran, M.: Valeurs propres et vecteurs propres en classification hiérarchique. R. A. I. R. O. Informatique Théorique 10, 39–46 (1976)Google Scholar
  12. 12.
    Gondran, M., Minoux, M.: Eigenvalues and eigenvectors in semimodules and their interpretation in graph theory. In: Proc. 9th Prog. Symp., pp. 133–148 (1976)Google Scholar
  13. 13.
    Gondran, M., Minoux, M.: Valeurs propres et vecteurs propres en théorie des graphes. Colloques Internationaux, pp. 181–183. CNRS, Paris (1978)Google Scholar
  14. 14.
    Gondran, M., Minoux, M.: Dioïds and semirings: Links to fuzzy sets and other applications. Fuzzy Sets Syst. 158, 1273–1294 (2007)CrossRefGoogle Scholar
  15. 15.
    Olsder, G.: Eigenvalues of dynamic max–min systems. In: Discrete Events Dynamic Systems, vol. 1, pp. 177–201. Kluwer, Dordrecht (1991)Google Scholar
  16. 16.
    Rashid, I.: Stability of discrete-time fuzzy systems, 99 pp. Doctoral thesis, University of Hradec Kralove, Czech Republic (2011)Google Scholar
  17. 17.
    Rashid, I., Gavalec, M., Sergeev, S.: Eigenspace of a three-dimensional max-Łukasiewicz fuzzy matrix. Kybernetika 48, 309–328 (2012)Google Scholar
  18. 18.
    Sanchez, E.: Resolution of eigen fuzzy sets equations. Fuzzy Sets Syst. 1, 69–74 (1978)CrossRefGoogle Scholar
  19. 19.
    Tan, Y.-J.: Eigenvalues and eigenvectors for matrices over distributive lattices. Lin. Algebra Appl. 283, 257–272 (1998)CrossRefGoogle Scholar
  20. 20.
    Tan, Y.-J.: On the powers of matrices over a distributive lattice. Lin. Algebra Appl. 336, 1–14 (2001)CrossRefGoogle Scholar
  21. 21.
    Zimmermann, U.: Linear and Combinatorial Optimization in Ordered Algebraic Structure. In: Ann. Discrete Math., vol. 10. North Holland, Amsterdam (1981)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of Hradec KraloveHradec KraloveCzech Republic
  2. 2.Silesian University in OpavaKarvinaCzech Republic
  3. 3.Charles University in PraguePragueCzech Republic

Personalised recommendations