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Journal of Applied Mathematics
Volume 2014, Article ID 202075, 10 pages
http://dx.doi.org/10.1155/2014/202075
Research Article

On Decompositions of Matrices over Distributive Lattices

1Department of Mathematics, Huizhou University, Huizhou, Guangdong 516007, China
2Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330027, China

Received 20 January 2014; Accepted 23 April 2014; Published 18 May 2014

Academic Editor: Jianming Zhan

Copyright © 2014 Yizhi Chen and Xianzhong Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a distributive lattice and ( , resp.) the semigroup (semiring, resp.) of ( , resp.) matrices over . In this paper, we show that if there is a subdirect embedding from distributive lattice to the direct product of distributive lattices , then there will be a corresponding subdirect embedding from the matrix semigroup (semiring , resp.) to semigroup (semiring , resp.). Further, it is proved that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the decomposition theorems of matrices over finite distributive lattices, chain semirings, fuzzy semirings, and so forth. Finally, as some applications, we present a method to calculate the indices and periods of the matrices over a distributive lattice and characterize the structures of idempotent and nilpotent matrices over it. We translate the characterizations of idempotent and nilpotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which also generalize the corresponding structures of idempotent and nilpotent matrices over general Boolean algebras, chain semirings, fuzzy semirings, and so forth.