Sufficient conditions for the ergodicity of fuzzy Markov chains

Analogous to the traditional Markov chains which have been studied extensively and have many successful applications, fuzzy Markov chains have been proposed for decision-making in an environment of uncertainty and imprecision for decades. It is known that results of fuzzy Markov chains depend on the transition matrix as well as the algebraic composition involved. In their study of max-min fuzzy Markov chains, Avrachenkov and Sanchez raised an open question for finding conditions to ensure the ergodicity of max-min fuzzy Markov chains. In this paper, we provide sufficient conditions for the ergodicity of both max-min and max-product fuzzy Markov chains. It is not surprising that such sufficient conditions are very different because of the max-min and max-product compositions.