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On n x n matrices over a finite distributive lattice

Author:
Chen, Yizhi  Zhao, Xianzhong  Yang, Lin  


Journal:
LINEAR & MULTILINEAR ALGEBRA


Issue Date:
2012


Abstract(summary):

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.


Page:
131---147


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