On the max-generalized mean powers of a fuzzy matrix

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.