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Eigenproblem in Max-Drast and Max-Łukasiewicz Algebra

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

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 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.

Keywords

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.

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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

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