Abstract
The eigenvectors of a fuzzy matrix correspond to steady states of a complex discrete-events system, characterized by the given transition matrix and fuzzy state vectors. The descriptions of the eigenspace for matrices in the max-Lukasiewicz algebra, max-min algebra, max-nilpotent-min algebra, max-product algebra and max-drast algebra have been presented in previous papers. In this paper, we investigate the monotone eigenvectors in a max-T algebra, list some particular properties of the monotone eigenvectors in max-Lukasiewicz algebra, max-min algebra, max-nilpotent-min algebra, max-product algebra and max-drast algebra, respectively, and illustrate the relations among eigenspaces in these algebras by some examples.
Highlights
The eigenproblem for a fuzzy matrix corresponds to finding a stable state of the complex discrete-events system described by the given transition matrix and fuzzy state vectors
The investigation of the eigenspace structure in fuzzy algebras is important for application
We investigate the eigenvectors in a max-T algebra, study monotone eigenvectors in max-nilpotent-min algebra, discuss the relation between the monotone eigenvectors in max-T algebra and max-drast algebra, and illustrate the relations among eigenspaces in these algebras by some examples
Summary
The eigenproblem for a fuzzy matrix corresponds to finding a stable state (or all stable states) of the complex discrete-events system described by the given transition matrix and fuzzy state vectors. A max-T fuzzy algebra is defined over the interval [0,1] and uses, instead of the conventional operations of addition and multiplication, the operations of maximum and one of the triangular norms, the so-called t-norm. These operations are extended in a natural way to the Cartesian products of vectors and matrices. Gavalec et al [2] [3] investigated the steady states of max-Łukasiewicz fuzzy systems and monotone interval eigenproblem in max-min algebra, Rashid et al [4] discussed the eigenspace structure of a max-product fuzzy matrix and Gavalec et al [5] studied the eigenspace structure of a max-drast fuzzy matrix. We investigate the eigenvectors in a max-T algebra, study monotone eigenvectors in max-nilpotent-min algebra, discuss the relation between the monotone eigenvectors in max-T algebra and max-drast algebra, and illustrate the relations among eigenspaces in these algebras by some examples
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