A congruence on the semiring of normal tropical matrices

We introduce and study a congruence ρ on the normal tropical matrix semiring MnN, which is relevant to the Kleene stars of normal tropical matrices. We prove that this congruence is a bisemilattice congruence and give an exact description of each ρ-class. In particular, we show that the ρ-class Eρ is an interval when E is a strongly regular normal tropical matrix. We also present a method using Floyd-Warshall algorithm to compute the greatest lower bound of Eρ in such case.