On nilpotent incline matrices

Inclines are additively idempotent semirings in which products are less than or equal to either factor. Thus they generalize Boolean algebra, fuzzy algebra and distributive lattice. This paper studies the nilpotent incline matrices in detail. It is proved that an incline matrix is nilpotent if and only if it has index and the zero vector is its unique standard eigenvector. The nilpotent matrices over an incline without nilpotent elements are characterized in terms of principal minors, main diagonals, nilpotent indices and adjoint matrices. Also some properties of the reduction of nilpotent matrices over an additively residuated incline without nilpotent elements are established. The results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references.