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Abstract. An exponent matrix is an n×n matrix A = (a ij ) over N 0 satisfying (1) a ii = 0 for all i = 1, . . . , n and (2) a ij + a jk ≥ a ik for all pairwise distinct i, j, k ∈ {1, . . . , n}. In the present paper we study the set E n of all non-negative n × n exponent matrices as an algebra with the operations ⊕ of component-wise maximum and ⊙ of component-wise addition. We provide a basis of the algebra (E n , ⊕, ⊙, 0) and give a row and a column decompositions of a matrix A ∈ E n with respect to this basis. This structure result determines all n × n tiled orders over a fixed discrete valuation ring. We also study automorphisms of E n with respect to each of the operations ⊕ and ⊙ and prove that Aut(E n , ⊙) = Aut(E n , ⊕) = Aut(E n , ⊙, ⊕, 0) ≃ S n × C 2 , n > 2.