The Transitive Closures of Matrices over Idempotent Semirings and its Applications*

with a broad background. Consider the problem to solve the algebraic path problem can be concluded to find the Kleene closure of the adjacency matrix, an algorithm of time complexity 3 ()O n to find the transitive closure of matrices over idempotent semiring is constructed, as well as the conditions applicable to it are provided. Compared with the Gauβ-Jordan elimination, this algorithm could extend the applicable range to certain semirings which do not have completeness and closeness, Thus, it has a wide application since it can provide a new method to solve the algebraic path problem over idempotent semirings which do not have completeness and closeness