{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:31Z","timestamp":1740134551578,"version":"3.37.3"},"reference-count":34,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11261021","11226287","S2012040007195","2013LYM0086"],"award-info":[{"award-number":["11261021","11226287","S2012040007195","2013LYM0086"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11261021","11226287","S2012040007195","2013LYM0086"],"award-info":[{"award-number":["11261021","11226287","S2012040007195","2013LYM0086"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003453","name":"Natural Science Foundation of Guangdong Province","doi-asserted-by":"crossref","award":["11261021","11226287","S2012040007195","2013LYM0086"],"award-info":[{"award-number":["11261021","11226287","S2012040007195","2013LYM0086"]}],"id":[{"id":"10.13039\/501100003453","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Outstanding Young Innovative Talent Training Project in Guangdong Universities","award":["11261021","11226287","S2012040007195","2013LYM0086"],"award-info":[{"award-number":["11261021","11226287","S2012040007195","2013LYM0086"]}]},{"name":"Key Discipline Foundation of Huizhou University","award":["11261021","11226287","S2012040007195","2013LYM0086"],"award-info":[{"award-number":["11261021","11226287","S2012040007195","2013LYM0086"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"LetL<\/mml:mi><\/mml:mrow><\/mml:math>be a distributive lattice andM<\/mml:mi><\/mml:mrow>n<\/mml:mi>,<\/mml:mo>q<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>(L<\/mml:mi>)<\/mml:mo><\/mml:math>(M<\/mml:mi><\/mml:mrow>n<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>L<\/mml:mi>)<\/mml:mo><\/mml:math>, resp.) the semigroup (semiring, resp.) ofn<\/mml:mi>\u00d7<\/mml:mo>q<\/mml:mi><\/mml:math>(n<\/mml:mi>\u00d7<\/mml:mo>n<\/mml:mi><\/mml:math>, resp.) matrices overL<\/mml:mi><\/mml:mrow><\/mml:math>. In this paper, we show that if there is a subdirect embedding from distributive latticeL<\/mml:mi><\/mml:mrow><\/mml:math>to the direct product\u220f<\/mml:mo>i<\/mml:mi>=<\/mml:mo>1<\/mml:mn><\/mml:mrow>m<\/mml:mi><\/mml:mrow><\/mml:msubsup>\u200d<\/mml:mi><\/mml:mrow><\/mml:mrow>L<\/mml:mi><\/mml:mrow>i<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math>of distributive latticesL<\/mml:mi><\/mml:mrow>1<\/mml:mn><\/mml:mrow><\/mml:msub>,<\/mml:mo>L<\/mml:mi><\/mml:mrow>2<\/mml:mn><\/mml:mrow><\/mml:msub>,<\/mml:mo>\u00a0<\/mml:mi>\u2026<\/mml:mo>,<\/mml:mo>L<\/mml:mi><\/mml:mrow>m<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math>, then there will be a corresponding subdirect embedding from the matrix semigroupM<\/mml:mi><\/mml:mrow>n<\/mml:mi>,<\/mml:mo>q<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>L<\/mml:mi>)<\/mml:mo><\/mml:math>(semiringM<\/mml:mi><\/mml:mrow>n<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>L<\/mml:mi>)<\/mml:mo><\/mml:math>, resp.) to semigroup\u220f<\/mml:mo>i<\/mml:mi>=<\/mml:mo>1<\/mml:mn><\/mml:mrow>m<\/mml:mi><\/mml:mrow><\/mml:msubsup>\u200d<\/mml:mi><\/mml:mrow><\/mml:mrow>M<\/mml:mi><\/mml:mrow>n<\/mml:mi>,<\/mml:mo>q<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>L<\/mml:mi><\/mml:mrow>i<\/mml:mi><\/mml:mrow><\/mml:msub>)<\/mml:mo><\/mml:math>(semiring\u220f<\/mml:mo>i<\/mml:mi>=<\/mml:mo>1<\/mml:mn><\/mml:mrow>m<\/mml:mi><\/mml:mrow><\/mml:msubsup>\u200d<\/mml:mi><\/mml:mrow><\/mml:mrow>M<\/mml:mi><\/mml:mrow>n<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>L<\/mml:mi><\/mml:mrow>i<\/mml:mi><\/mml:mrow><\/mml:msub>)<\/mml:mo><\/mml:math>, 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.<\/jats:p>","DOI":"10.1155\/2014\/202075","type":"journal-article","created":{"date-parts":[[2014,5,18]],"date-time":"2014-05-18T21:02:36Z","timestamp":1400446956000},"page":"1-10","source":"Crossref","is-referenced-by-count":0,"title":["On Decompositions of Matrices over Distributive Lattices"],"prefix":"10.1155","volume":"2014","author":[{"given":"Yizhi","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Huizhou University, Huizhou, Guangdong 516007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2809-3400","authenticated-orcid":true,"given":"Xianzhong","family":"Zhao","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330027, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1017\/S0013091500016643"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(96)00364-3"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(92)90448-J"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2008.06.031"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(92)90394-P"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1080\/03081087.2011.574626"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.2307\/1969503"},{"key":"9"},{"key":"10","first-page":"1","volume":"2","year":"2004","journal-title":"International Journal of Computational Cognition"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005999"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0255(95)00283-9"},{"key":"13","first-page":"477","volume":"7","year":"1964","journal-title":"Information and Computation"},{"issue":"3","key":"15","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/0020-0255(82)90027-5","volume":"27","year":"1982","journal-title":"Information Sciences"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.4134\/BKMS.2007.44.4.721"},{"volume":"70","year":"1982","series-title":"Monographs and Textbooks in Pure and Applied Mathematics","key":"18"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(92)90308-W"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1080\/03081089308818200"},{"issue":"1","key":"21","first-page":"52","volume":"3","year":"1989","journal-title":"Fuzzy Systems and Mathematics"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(02)00587-6"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2003.08.006"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.4134\/JKMS.2010.47.4.805"},{"issue":"1","key":"25","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1070\/RM2009v064n01ABEH004604","volume":"64","year":"2009","journal-title":"Russian Mathematical Surveys"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(84)90070-X"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.4134\/JKMS.2007.44.1.169"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2008.02.015"},{"key":"29","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(00)00168-3"},{"key":"30","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(01)00246-9"},{"key":"31","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2004.06.009"},{"key":"32","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2004.11.016"},{"key":"33","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2008.03.004"},{"key":"34","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(98)00270-X"},{"year":"2000","series-title":"The Millennium Edition","key":"6"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-9333-5"},{"volume":"12","year":"1995","series-title":"London Mathematical Society Monographs. New Series","key":"16"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/202075.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/202075.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/202075.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,11]],"date-time":"2020-05-11T23:29:36Z","timestamp":1589239776000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/202075\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":34,"alternative-id":["202075","202075"],"URL":"https:\/\/doi.org\/10.1155\/2014\/202075","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2014]]}}}