The max-plus algebra of exponent matrices of tiled orders

Abstract

An exponent matrix is an $n\times n$ matrix $A=(a_{ij})$ over ${\mathbb N}^0$ satisfying (1) $a_{ii}=0$ for all $i=1,\ldots, n$ and (2) $a_{ij}+a_{jk}\geq a_{ik}$ for all pairwise distinct $i,j,k\in\{1,\dots, n\}$. In the present paper we study the set ${\mathcal E}_n$ of all non-negative $n\times n$ exponent matrices as an algebra with the operations $\oplus$ of component-wise maximum and $\odot$ of component-wise addition. We provide a basis of the algebra $({\mathcal E}_n, \oplus, \odot,0)$ and give a row and a column decompositions of a matrix $A\in {\mathcal E}_n$ with respect to this basis. This structure result determines all $n\times n$ tiled orders over a fixed discrete valuation ring. We also study automorphisms of ${\mathcal E}_n$ with respect to each of the operations $\oplus$ and $\odot$ and prove that ${\rm Aut}(\mathcal{E}_n,\, \odot ) = {\rm Aut}(\mathcal{E}_n,\, \oplus ) = {\rm Aut}(\mathcal{E}_n,\, \odot ,\oplus ,0) \simeq {\mathcal{S}}_n \times C_2,$$n>2.$