On the transitive matrices over distributive lattices

A matrix is called a lattice matrix if its elements belong to a distributive lattice. For a lattice matrix A of order n, if there exists an n × n permutation matrix P such that F = PAPT = (fij) satisfies fij ≮ fji for i > j, then F is called a canonical form of A. In this paper, the transitivity of powers and the transitive closure of a lattice matrix are studied, and the convergence of powers of transitive lattice matrices is considered. Also, the problem of the canonical form of a transitive lattice matrix is further discussed.