Eigenproblem in Max-Drast and Max-Łukasiewicz Algebra

Martin Gavalec, Jaroslav Ramík and Karel Zimmermann
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Martin Gavalec: University of Hradec Kralove
Jaroslav Ramík: Silesian University in Opava
Karel Zimmermann: Charles University in Prague

Chapter Chapter 6 in Decision Making and Optimization, 2015, pp 183-221 from Springer

Abstract: 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.

Keywords: Fuzzy Transition Matrix; Triangular Norms; Fuzzy Algebra; Eigenspace Structure; General Eigenproblem (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-08323-0_6

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