IJPAM: Volume 75, No. 2 (2012)

DECOMPOSITIONS OF MATRICES OVER A FINITE CHAIN

Yizhi Chen$^1$, Haiqing Wang$^2$, Hui Luo$^3$
Department of Mathematics
Huizhou University
Huizhou, Guangdong, 516007, P.R. CHINA


Abstract. Let $L=\{0, 1, ..., l\}$ denote a finite chain, $M_{m,n}(L)$ be the additive semigroup of all the $m\times n$ matrices over $L$. In this paper, we firstly give some subdirect decompositions of a finite chain $L$, and then show that if there is a subdirect embedding from $L$ to the direct product $\prod_{i=1}^{h}L_i$ of subchains $L_1, L_2,\,\cdots,L_h$, then there will be a corresponding subdirect embedding from the semigroup $M_{m,n}(L)$ to semigroup $\prod_{i=1}^{h}\text{M}_{m,n}(L_i)$. Based on the above results, it is also proved that a matrix $A\in M_{m,n}(L)$ can be decomposed into the sum of matrices over some special subchains of $L$ which generalizes and extends the corresponding results obtained by [1].

Received: June 14, 2011

AMS Subject Classification: 20M10, 15A09, 16Y60

Key Words and Phrases: decomposition, matrix, finite chain, subdirect product, semigroup, semiring

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 75
Issue: 2