The max-plus algebra of exponent matrices of tiled orders

Abstract

An exponent matrix is an n×n matrix A=(aij) over N0 satisfying (1) aii=0 for all i=1,…,n and (2) aij+ajk≥aik for all pairwise distinct i,j,k∈{1,…,n}. In the present paper we study the set En 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 (En,⊕,⊙,0) and give a row and a column decompositions of a matrix A∈En with respect to this basis. This structure result determines all n×n-tiled orders over a fixed discrete valuation domain. We also study automorphisms of En with respect to each of the operations ⊕ and ⊙ and prove that Aut(En,⊕,⊙,0)≅Aut(En,⊕)≅Aut(En,⊙)≅Sn×C2, n>2.