On the max-nilpotent t-norm powers of a fuzzy matrix

Abstract

Fuzzy matrices have been proposed to represent fuzzy relations in finite universes. Various studies have evaluated the powers of a fuzzy matrix with max–min/max-product/max Archimedean t-norm/max t-norm/max-arithmetic mean compositions, indicating that the limiting behavior of the powers of a fuzzy matrix depends on its composition. In this paper, max-nilpotent composition is considered for the fuzzy relations. We demonstrate that the max-nilpotent powers of a fuzzy matrix either converge or oscillate in a finite period. Moreover, the max-nilpotent t-norm powers of a fuzzy matrix A are p-periodic if and only if the powers of an associated Boolean matrix Ac are p-periodic. Finally, necessary and sufficient conditions are proposed for nilpotent fuzzy matrices that exhibit max-nilpotent composition.