The semiring of matrices over a finite chain

Let LmLm denote the chain {0,1,2,…,m-1}{0,1,2,…,m-1} with the usual ordering and Mn(Lm)Mn(Lm) the matrix semiring of all n×nn×n matrices with elements in LmLm. We firstly introduce some order-preserving semiring homomorphisms from Mn(Lm)Mn(Lm) to M(Lk)M(Lk). By using these homomorphisms, we show that a matrix over the finite chain LmLm can be decomposed into the sum of some matrices over the finite chain LkLk, where k