On n x n matrices over a finite distributive lattice

In this article, we study the n x n matrices over a finite distributive lattice L. By using join irreducible elements in L, we first give some concrete ways to decompose L into a subdirect product of some chains. Also, it is showed that by a subdirect embedding from semiring R to the direct product Pi(m)(i=1) R-i of semirings R-1, R-2, ... , R-m, we can give a corresponding subdirect embedding from the matrix semiring M-n(R) to semiring Pi(m)(i=1) M-n(R-i). Based on the above results, it is proved that a square matrix over a finite distributive lattice L can be decomposed into the sum of matrices over some special subchains of L. This generalizes and extends the corresponding results obtained by Fan [Z.T. Fan, The Theory and Applications of Fuzzy Matrices, Science Publication, Beijing, 2006 (in Chinese)] and by Zhao et al. [X.Z. Zhao, Y.B. Jun, and F. Ren, The semiring of matrices over a finite chain, Inform. Sci. 178 (2008), pp. 3443-3450]. As some applications, we present a method to calculate the indices and periods of the matrices over a finite distributive lattice, and characterize the idempotent and nilpotent matrices over a finite distributive lattice. Also, we discuss Green's relations on the multiplicative semigroup of semiring of matrices over a finite distributive lattice.