The semiring of matrices over a finite chain

Let image denote the chain image with the usual ordering and image the matrix semiring of all image matrices with elements in image. We firstly introduce some order-preserving semiring homomorphisms from image to image. By using these homomorphisms, we show that a matrix over the finite chain image can be decomposed into the sum of some matrices over the finite chain image, where image. As a result, cut matrices decomposition theorem of a fuzzy matrix (Theorem 4 in [Z.T. Fan, Q.S. Cheng, A survey on the powers of fuzzy matrices and FBAMs, International Journal of Computational Cognition 2 (2004) 1–25 (invited paper)]) is generalized and extended. Further, we study the index and periodicity of a matrix over a finite chain and get some new results. On the other hand, we introduce a semiring embedding mapping from the semiring image to the direct product of the h copies of the semiring image and discuss Green’s relations on the multiplicative semigroup of the semiring image. We think that some results obtained in this paper is useful for the study of fuzzy matrices.