The semiring of matrices over a finite chain

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• 作者：Xianzhong Zhao ; Young Bae Jun ; Fang Ren
• 关键词：Semiring ; Matrix ; Fuzzy matrix ; Index ; Period ; Chain ; Homomorphism
• 刊名：Information Sciences
• 出版年：2008
• 出版时间：1 September 2008
• 年：2008
• 卷：178
• 期：17
• 页码：3443-3450
• 全文大小：205 K

文摘

Let L_m denote the chain {0,1,2,…,m-1} with the usual ordering and M_n(L_m) the matrix semiring of all n×n matrices with elements in L_m. We firstly introduce some order-preserving semiring homomorphisms from M_n(L_m) to M(L_k). By using these homomorphisms, we show that a matrix over the finite chain L_m can be decomposed into the sum of some matrices over the finite chain L_k, where k<m. 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 M_n(L_m) to the direct product of the h copies of the semiring M_n(L_k) and discuss Green’s relations on the multiplicative semigroup of the semiring M_n(L_m). We think that some results obtained in this paper is useful for the study of fuzzy matrices.