{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:54:11Z","timestamp":1753883651551,"version":"3.41.2"},"reference-count":15,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,8,14]],"date-time":"2012-08-14T00:00:00Z","timestamp":1344902400000},"content-version":"vor","delay-in-days":226,"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":["61179026","ZXH2012K003"],"award-info":[{"award-number":["61179026","ZXH2012K003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["61179026","ZXH2012K003"],"award-info":[{"award-number":["61179026","ZXH2012K003"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>Let  be a (2<jats:italic>\u03bd<\/jats:italic> + <jats:italic>\u03b4<\/jats:italic> + <jats:italic>l<\/jats:italic>)\u2010dimensional vector space over the finite field <jats:italic>\ud835\udd3d<\/jats:italic><jats:sub><jats:italic>q<\/jats:italic><\/jats:sub>. In this paper we assume that <jats:italic>\ud835\udd3d<\/jats:italic><jats:sub><jats:italic>q<\/jats:italic><\/jats:sub> is a finite field of odd characteristic, and <jats:italic>O<\/jats:italic><jats:sub>2<jats:italic>\u03bd<\/jats:italic>+<jats:italic>\u03b4<\/jats:italic>+<jats:italic>l<\/jats:italic>,<jats:italic>\u2009\u2009<\/jats:italic>\u0394<\/jats:sub>(<jats:italic>\ud835\udd3d<\/jats:italic><jats:sub><jats:italic>q<\/jats:italic><\/jats:sub>) the singular orthogonal groups of degree 2<jats:italic>\u03bd<\/jats:italic> + <jats:italic>\u03b4<\/jats:italic> + <jats:italic>l<\/jats:italic> over <jats:italic>\ud835\udd3d<\/jats:italic><jats:sub><jats:italic>q<\/jats:italic><\/jats:sub>. Let <jats:italic>\u2133<\/jats:italic> be any orbit of subspaces under <jats:italic>O<\/jats:italic><jats:sub>2<jats:italic>\u03bd<\/jats:italic>+<jats:italic>\u03b4<\/jats:italic>+<jats:italic>l<\/jats:italic>,<jats:italic>\u2009\u2009<\/jats:italic>\u0394<\/jats:sub>(<jats:italic>\ud835\udd3d<\/jats:italic><jats:sub><jats:italic>q<\/jats:italic><\/jats:sub>). Denote by <jats:italic>\u2112<\/jats:italic> the set of subspaces which are intersections of subspaces in <jats:italic>\u2133<\/jats:italic>, where we make the convention that the intersection of an empty set of subspaces of  is assumed to be . By ordering <jats:italic>\u2112<\/jats:italic> by ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these lattices <jats:italic>\u2112<\/jats:italic> are geometric lattices.<\/jats:p>","DOI":"10.1155\/2012\/387132","type":"journal-article","created":{"date-parts":[[2012,8,14]],"date-time":"2012-08-14T21:00:36Z","timestamp":1344978036000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lattices Generated by Orbits of Subspaces under Finite Singular Orthogonal Groups II"],"prefix":"10.1155","volume":"2012","author":[{"given":"You","family":"Gao","sequence":"first","affiliation":[]},{"given":"XinZhi","family":"Fu","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,8,14]]},"reference":[{"key":"e_1_2_6_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2007.09.046"},{"key":"e_1_2_6_2_2","doi-asserted-by":"publisher","DOI":"10.1080\/00927870500542697"},{"key":"e_1_2_6_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2007.06.004"},{"key":"e_1_2_6_4_2","doi-asserted-by":"publisher","DOI":"10.1080\/00927879208824395"},{"key":"e_1_2_6_5_2","doi-asserted-by":"publisher","DOI":"10.1080\/00927879208824485"},{"key":"e_1_2_6_6_2","doi-asserted-by":"publisher","DOI":"10.1080\/00927879308824681"},{"key":"e_1_2_6_7_2","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.2001.8819"},{"volume-title":"Lattices Generated by Orbits of Subspaces under Finite Classical Groups","year":"2002","author":"Wan Z.","key":"e_1_2_6_8_2"},{"key":"e_1_2_6_9_2","doi-asserted-by":"publisher","DOI":"10.1081\/AGB-120021900"},{"key":"e_1_2_6_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(02)00674-2"},{"key":"e_1_2_6_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2009.05.011"},{"key":"e_1_2_6_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2009.01.003"},{"key":"e_1_2_6_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2010.06.004"},{"key":"e_1_2_6_14_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-6666-3"},{"volume-title":"Geometry of Classical Groups Over Finite Fields","year":"2002","author":"Wan Z.","key":"e_1_2_6_15_2"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/387132.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/387132.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2012\/387132","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T07:04:44Z","timestamp":1718089484000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2012\/387132"}},"subtitle":[],"editor":[{"given":"Ch","family":"Tsitouras","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1155\/2012\/387132"],"URL":"https:\/\/doi.org\/10.1155\/2012\/387132","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2012,1]]},"assertion":[{"value":"2012-03-28","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-06-15","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-08-14","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"387132"}}