## Abstract

Fuzzy matrices have been proposed to represent fuzzy relations on finite universes. Since Thomason's paper in 1977 showing that the max-min powers of a fuzzy matrix either converge or oscillate with a finite period, conditions for limiting behavior of powers of a fuzzy matrix have been studied. It turns out that the limiting behavior depends on the algebraic operations employed, which usually in the literature include max-min/max-product/max-Archimedean t-norm/max-t-norm/max-arithmetic mean operations, respectively. In this paper, we consider the max-generalized mean powers of a fuzzy matrix which is an extension of the max-arithmetic mean operation. We show that the powers of such fuzzy matrices are always convergent. As an application, we consider fuzzy Markov chains with the max-generalized mean operations for the fuzzy transition matrix. Our results imply that these fuzzy Markov chains are always ergodic and robust with respect to small perturbations of the transition matrices.

Original language | English |
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Pages (from-to) | 750-762 |

Number of pages | 13 |

Journal | Fuzzy Sets and Systems |

Volume | 161 |

Issue number | 5 |

DOIs | |

State | Published - 01 03 2010 |

Externally published | Yes |

## Keywords

- Convergence
- Ergodicity
- Fuzzy Markov chains
- Max-arithmetic mean composition
- Max-generalized mean composition
- Powers of a fuzzy matrix