Indices and periods of incline matrices

Abstract

Inclines are the additively idempotent semirings in which products are less than or equal to factors. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. And the Boolean matrices, the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices (i.e., the matrices over inclines). This paper studies the power sequence of incline matrices in detail. A necessary and sufficient condition for the incline matrix to have index is given and the indices of some incline matrices with indices are estimated. It is proved that the period of n× n incline matrix with index is a divisor of [ n] and that the set of periods of n× n incline matrices with indices is not bounded from above in the sense of a power of n for all n. An equivalent condition and some sufficient conditions for the incline matrix to converge in finite steps are established. The stability of the orbits of an incline matrix is considered and a theorem in [Fuzzy Sets and Systems 81 (1996) 227] is pointed out being false. The results in the present paper include some previous results in the literatures which were obtained for the Boolean matrices, the fuzzy matrices and the lattice matrices among their special cases.