{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T14:11:24Z","timestamp":1777644684910,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"3","license":[{"start":{"date-parts":[[2012,1,1]],"date-time":"2012-01-01T00:00:00Z","timestamp":1325376000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Fundamenta Informaticae"],"published-print":{"date-parts":[[2012,9]]},"abstract":"<jats:p>\n                    Inspired by the bimodule matrix problem technique and various classification problems in poset representation theory, finite groups and algebras, we study the action of Belitskii algorithm on a class of square n by n block matrices M with coefficients in a field K. One of the main aims is to reduce M to its special canonical form M\n                    <jats:sup>\u221e<\/jats:sup>\n                    with respect to the conjugation by elementary transformations defined by a class of matrices chosen in a subalgebra of the full matrix algebra $\\mathbb{M}_n$(K). The algorithm can be successfully applied in the study of indecomposable linear representations of finite posets by a computer search using numeric and symbolic computation. We mainly study the case when the di-graph (quiver) associated to the output matrix M\n                    <jats:sup>\u221e<\/jats:sup>\n                    of the algorithm is a disjoint union of trees. We show that exceptional representations of any finite poset are determined by tree matrices. This generalizes a theorem of C.M. Ringel proved for linear representations of di-graphs.\n                  <\/jats:p>","DOI":"10.3233\/fi-2012-713","type":"journal-article","created":{"date-parts":[[2019,12,3]],"date-time":"2019-12-03T00:09:04Z","timestamp":1575331744000},"page":"253-279","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Tree Matrices and a Matrix Reduction Algorithm of Belitskii"],"prefix":"10.1177","volume":"118","author":[{"given":"Marcin","family":"Grzecza","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12\/18, 87-100 Toru\u0144, Poland, grzeczny@mat.umk.pl; skasjan@mat.umk.pl; amroz@mat.umk.pl"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stanis\u0142aw","family":"Kasjan","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12\/18, 87-100 Toru\u0144, Poland, grzeczny@mat.umk.pl; skasjan@mat.umk.pl; amroz@mat.umk.pl"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrzej","family":"Mr\u00f3z","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12\/18, 87-100 Toru\u0144, Poland, grzeczny@mat.umk.pl; skasjan@mat.umk.pl; amroz@mat.umk.pl"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2012,1,1]]},"container-title":["Fundamenta Informaticae"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/FI-2012-713","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/FI-2012-713","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T06:30:06Z","timestamp":1777444206000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/FI-2012-713"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,1]]},"references-count":0,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,9]]}},"alternative-id":["10.3233\/FI-2012-713"],"URL":"https:\/\/doi.org\/10.3233\/fi-2012-713","relation":{},"ISSN":["0169-2968","1875-8681"],"issn-type":[{"value":"0169-2968","type":"print"},{"value":"1875-8681","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,1,1]]}}}