On nilpotent matrices over distributive lattices

In this paper, nilpotent matrices over distributive lattices are considered. Some properties and characterizations for nilpotent matrices are established and in particular, a necessary and sufficient condition for an n×n nilpotent matrix to have the nilpotent index n is given. Also, some properties of the reduction of a nilpotent matrix are obtained. Partial results obtained in this paper are results on nilpotent fuzzy matrices by H. Hashimoto.