Convergence of powers for a fuzzy matrix with convex combination of max-min and max-arithmetic mean operations

Fuzzy matrices have been proposed to represent fuzzy relations on finite universes. Since Thomason's paper in 1977 showing that powers of a max-min 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 includes max-min/max-product/max-Archimedean t-norm/max t-norm/max-arithmetic mean operations, respectively. In this paper, we consider the powers of a fuzzy matrix with convex combination of max-min and max-arithmetic mean operations. We show that the powers of such a fuzzy matrix are always convergent.