On Decompositions of Matrices over Distributive Lattices
Let L be a distributive lattice and Mn,q (L)(Mn(L), resp.) the semigroup (semiring, resp.) of n × q (n × n, resp.) matrices over L. In this paper, we show that if there is a subdirect embedding from distributive lattice L to the direct product ∏i=1mLi of distributive lattices L1,L2, …,Lm, then ther...
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Wiley
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/202075 |
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| author | Yizhi Chen Xianzhong Zhao |
| author_facet | Yizhi Chen Xianzhong Zhao |
| author_sort | Yizhi Chen |
| collection | DOAJ |
| description | Let L be a distributive lattice and Mn,q (L)(Mn(L), resp.) the semigroup (semiring, resp.) of n × q (n × n, resp.) matrices over L. In this paper, we show that if there is a subdirect embedding from distributive lattice L to the direct product ∏i=1mLi of distributive lattices L1,L2, …,Lm, then there will be a corresponding subdirect embedding from the matrix semigroup Mn,q(L) (semiring Mn(L), resp.) to semigroup ∏i=1mMn,q(Li) (semiring ∏i=1mMn(Li), 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. |
| format | Article |
| id | doaj-art-083374acb9bc481fbf98f8f89d1a7f74 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-083374acb9bc481fbf98f8f89d1a7f742025-02-03T07:26:16ZengWileyJournal of Applied Mathematics1110-757X1687-00422014-01-01201410.1155/2014/202075202075On Decompositions of Matrices over Distributive LatticesYizhi Chen0Xianzhong Zhao1Department of Mathematics, Huizhou University, Huizhou, Guangdong 516007, ChinaInstitute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330027, ChinaLet L be a distributive lattice and Mn,q (L)(Mn(L), resp.) the semigroup (semiring, resp.) of n × q (n × n, resp.) matrices over L. In this paper, we show that if there is a subdirect embedding from distributive lattice L to the direct product ∏i=1mLi of distributive lattices L1,L2, …,Lm, then there will be a corresponding subdirect embedding from the matrix semigroup Mn,q(L) (semiring Mn(L), resp.) to semigroup ∏i=1mMn,q(Li) (semiring ∏i=1mMn(Li), 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.http://dx.doi.org/10.1155/2014/202075 |
| spellingShingle | Yizhi Chen Xianzhong Zhao On Decompositions of Matrices over Distributive Lattices Journal of Applied Mathematics |
| title | On Decompositions of Matrices over Distributive Lattices |
| title_full | On Decompositions of Matrices over Distributive Lattices |
| title_fullStr | On Decompositions of Matrices over Distributive Lattices |
| title_full_unstemmed | On Decompositions of Matrices over Distributive Lattices |
| title_short | On Decompositions of Matrices over Distributive Lattices |
| title_sort | on decompositions of matrices over distributive lattices |
| url | http://dx.doi.org/10.1155/2014/202075 |
| work_keys_str_mv | AT yizhichen ondecompositionsofmatricesoverdistributivelattices AT xianzhongzhao ondecompositionsofmatricesoverdistributivelattices |