Standard eigenvectors of incline matrices Original Research Article

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. This paper studies the standard eigenvectors of incline matrices (i.e., matrices over inclines) in detail. As the main results, for an incline matrix with index and for a lattice matrix, the structures of all the standard eigenvectors are given and the greatest standard eigenvector is determined. Also it is proved that for an incline matrix with index, the set of all the standard eigenvectors of it coincides with the set of all the standard eigenvectors of its transitive closure. The power convergence of an incline matrix with index is characterized in terms of the sets of standard eigenvectors. The results in the present paper include some previous results for the Boolean matrices, the fuzzy matrices and the lattice matrices among their special cases.