{"id":"https://openalex.org/W3010597046","doi":"https://doi.org/10.3390/sym13071125","title":"A Note on Decomposable and Reducible Integer Matrices","display_name":"A Note on Decomposable and Reducible Integer Matrices","publication_year":2021,"publication_date":"2021-06-24","ids":{"openalex":"https://openalex.org/W3010597046","doi":"https://doi.org/10.3390/sym13071125","mag":"3010597046"},"language":"en","primary_location":{"id":"doi:10.3390/sym13071125","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13071125","pdf_url":"https://www.mdpi.com/2073-8994/13/7/1125/pdf?version=1624548049","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/13/7/1125/pdf?version=1624548049","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051148586","display_name":"Carlos Mariju\u00e1n","orcid":"https://orcid.org/0000-0002-0241-8158"},"institutions":[{"id":"https://openalex.org/I108103353","display_name":"Universidad de Valladolid","ror":"https://ror.org/01fvbaw18","country_code":"ES","type":"education","lineage":["https://openalex.org/I108103353"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Carlos Mariju\u00e1n","raw_affiliation_strings":["Departamento de Matem\u00e1tica Aplicada/IMUVA, Universidad de Valladolid, E-47002 Valladolid, Spain"],"raw_orcid":"https://orcid.org/0000-0002-0241-8158","affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1tica Aplicada/IMUVA, Universidad de Valladolid, E-47002 Valladolid, Spain","institution_ids":["https://openalex.org/I108103353"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029519195","display_name":"Ignacio Ojeda","orcid":"https://orcid.org/0000-0003-3173-5934"},"institutions":[{"id":"https://openalex.org/I80606768","display_name":"Universidad de Extremadura","ror":"https://ror.org/0174shg90","country_code":"ES","type":"education","lineage":["https://openalex.org/I80606768"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Ignacio Ojeda","raw_affiliation_strings":["Departamento de Matem\u00e1ticas/IMUEX, Universidad de Extremadura, E-06071 Badajoz, Spain"],"raw_orcid":"https://orcid.org/0000-0003-3173-5934","affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1ticas/IMUEX, Universidad de Extremadura, E-06071 Badajoz, Spain","institution_ids":["https://openalex.org/I80606768"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025784427","display_name":"A. Vigneron-Tenorio","orcid":"https://orcid.org/0000-0003-4583-0852"},"institutions":[{"id":"https://openalex.org/I39147953","display_name":"Universidad de C\u00e1diz","ror":"https://ror.org/04mxxkb11","country_code":"ES","type":"education","lineage":["https://openalex.org/I39147953"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Alberto Vigneron-Tenorio","raw_affiliation_strings":["Departamento de Matem\u00e1ticas/INDESS, Universidad de C\u00e1diz, E-11406 Jerez de la Frontera, Spain"],"raw_orcid":"https://orcid.org/0000-0003-4583-0852","affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1ticas/INDESS, Universidad de C\u00e1diz, E-11406 Jerez de la Frontera, Spain","institution_ids":["https://openalex.org/I39147953"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5025784427"],"corresponding_institution_ids":["https://openalex.org/I39147953"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.04418847,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"13","issue":"7","first_page":"1125","last_page":"1125"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.7201589941978455},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5389737486839294},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5294729471206665},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2544080317020416}],"concepts":[{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.7201589941978455},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5389737486839294},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5294729471206665},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2544080317020416},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/sym13071125","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13071125","pdf_url":"https://www.mdpi.com/2073-8994/13/7/1125/pdf?version=1624548049","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2003.01453","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2003.01453","pdf_url":"https://arxiv.org/pdf/2003.01453","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:doaj.org/article:48e47c670eed4b558e7f7668545041cd","is_oa":true,"landing_page_url":"https://doaj.org/article/48e47c670eed4b558e7f7668545041cd","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 13, Iss 7, p 1125 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/7/1125/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13071125","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"},{"id":"pmh:oai:rodin.uca.es:10498/25643","is_oa":true,"landing_page_url":"http://hdl.handle.net/10498/25643","pdf_url":null,"source":{"id":"https://openalex.org/S4306400506","display_name":"RODIN (Universidad de C\u00e1diz)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I39147953","host_organization_name":"Universidad de C\u00e1diz","host_organization_lineage":["https://openalex.org/I39147953"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry 2021, 13(7), 1125","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.3390/sym13071125","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13071125","pdf_url":"https://www.mdpi.com/2073-8994/13/7/1125/pdf?version=1624548049","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2338927219","display_name":null,"funder_award_id":"PGC2018-096446-B-C21","funder_id":"https://openalex.org/F4320322930","funder_display_name":"Ministerio de Ciencia e Innovaci\u00f3n"},{"id":"https://openalex.org/G3913480368","display_name":null,"funder_award_id":"FQM-024","funder_id":"https://openalex.org/F4320328352","funder_display_name":"Junta de Extremadura"},{"id":"https://openalex.org/G5238818737","display_name":null,"funder_award_id":"MTM2017-84890-P","funder_id":"https://openalex.org/F4320322930","funder_display_name":"Ministerio de Ciencia e Innovaci\u00f3n"},{"id":"https://openalex.org/G8550076459","display_name":null,"funder_award_id":"FQM-366","funder_id":"https://openalex.org/F4320321562","funder_display_name":"Consejer\u00eda de Econom\u00eda, Innovaci\u00f3n, Ciencia y Empleo, Junta de Andaluc\u00eda"}],"funders":[{"id":"https://openalex.org/F4320321562","display_name":"Consejer\u00eda de Econom\u00eda, Innovaci\u00f3n, Ciencia y Empleo, Junta de Andaluc\u00eda","ror":"https://ror.org/03kak1729"},{"id":"https://openalex.org/F4320322930","display_name":"Ministerio de Ciencia e Innovaci\u00f3n","ror":"https://ror.org/034900433"},{"id":"https://openalex.org/F4320328352","display_name":"Junta de Extremadura","ror":"https://ror.org/01df4mv68"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3010597046.pdf","grobid_xml":"https://content.openalex.org/works/W3010597046.grobid-xml"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W651067426","https://openalex.org/W1487444668","https://openalex.org/W1537258143","https://openalex.org/W2007012222","https://openalex.org/W2023002777","https://openalex.org/W2043888222","https://openalex.org/W2130870229","https://openalex.org/W2144267444","https://openalex.org/W2171712069","https://openalex.org/W4205502224","https://openalex.org/W4301621763"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027","https://openalex.org/W2962838298","https://openalex.org/W2600246793"],"abstract_inverted_index":{"We":[0],"propose":[1],"necessary":[2],"and":[3,42],"sufficient":[4],"conditions":[5],"for":[6],"an":[7],"integer":[8,23,29],"matrix":[9,30],"to":[10,21],"be":[11,34],"decomposable":[12],"in":[13],"terms":[14],"of":[15,48],"its":[16],"Hermite":[17],"normal":[18],"form.":[19],"Specifically,":[20],"each":[22],"matrix,":[24],"we":[25],"associate":[26],"a":[27],"symmetric":[28],"whose":[31],"reducibility":[32],"can":[33],"efficiently":[35],"determined":[36],"by":[37],"elementary":[38],"linear":[39],"algebra":[40],"techniques,":[41],"which":[43],"completely":[44],"determines":[45],"the":[46,49],"decomposability":[47],"first":[50],"one.":[51]},"counts_by_year":[],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2020-03-13T00:00:00"}