The max-plus algebra of exponent matrices of tiled orders

An exponent matrix is an n x 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 is an element of{1, ... , n}. In the present paper we study the set epsilon(n) of all non-negative n x n exponent matrices as an algebra with the operations circle plus of component-wise maximum and circle dot of component-wise addition. We provide a basis of the algebra (epsilon(n), circle plus, circle dot, 0) and give a row and a column decompositions of a matrix A is an element of epsilon(n) with respect to this basis. This structure result determines all n x n-tiled orders over a fixed discrete valuation domain. We also study automorphisms of epsilon(n) with respect to each of the operations circle plus and circle dot and prove that Aut(epsilon(n), circle plus, circle dot, 0) congruent to Aut(epsilon(n), circle plus) congruent to Aut(epsilon(n), circle dot) congruent to S-n X C-2, n > 2. (C) 2017 Elsevier Inc. All rights reserved.