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Eigenproblem in Max-Drast and Max-Łukasiewicz Algebra

In: Decision Making and Optimization

Author

Listed:
  • Martin Gavalec

    (University of Hradec Kralove)

  • Jaroslav Ramík

    (Silesian University in Opava)

  • Karel Zimmermann

    (Charles University in Prague)

Abstract

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 Algebra max-T 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 Algebra max-drast , in which the least t-norm T (often called the drastic norm) is used, and max-Lukasiewicz algebra Algebra max-Lukasiewicz with Łukasiewicz t-norm L. For both of these max-T algebras the necessary and sufficient conditions are presented under which the monotone eigenspace 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.

Suggested Citation

  • Martin Gavalec & Jaroslav Ramík & Karel Zimmermann, 2015. "Eigenproblem in Max-Drast and Max-Łukasiewicz Algebra," Lecture Notes in Economics and Mathematical Systems, in: Decision Making and Optimization, edition 127, chapter 0, pages 183-221, Springer.
    • Handle: RePEc:spr:lnechp:978-3-319-08323-0_6
    DOI: 10.1007/978-3-319-08323-0_6
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