The max-plus algebra of exponent matrices of tiled orders

An exponent matrix is an $n\times n$ matrix $A=(a_{ij})$ over ${\mathbb{N}}^0$ satisfying (1) $a_{ii}=0$ for all $i=1,\dots ,n$ and (2) $a_{ij}+a_{jk}\geq a_{ik}$forallpairwisedistinct$i,j,k\in{1,\dots, n}$. 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.$