{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:06:29Z","timestamp":1760234789185,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,6,24]],"date-time":"2021-06-24T00:00:00Z","timestamp":1624492800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["PGC2018-096446-B-C21","MTM2017-84890-P"],"award-info":[{"award-number":["PGC2018-096446-B-C21","MTM2017-84890-P"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002878","name":"Consejer\u00eda de Econom\u00eda, Innovaci\u00f3n, Ciencia y Empleo, Junta de Andaluc\u00eda","doi-asserted-by":"publisher","award":["FQM-366"],"award-info":[{"award-number":["FQM-366"]}],"id":[{"id":"10.13039\/501100002878","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100014181","name":"Junta de Extremadura","doi-asserted-by":"publisher","award":["FQM-024"],"award-info":[{"award-number":["FQM-024"]}],"id":[{"id":"10.13039\/501100014181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We propose necessary and sufficient conditions for an integer matrix to be decomposable in terms of its Hermite normal form. Specifically, to each integer matrix, we associate a symmetric integer matrix whose reducibility can be efficiently determined by elementary linear algebra techniques, and which completely determines the decomposability of the first one.<\/jats:p>","DOI":"10.3390\/sym13071125","type":"journal-article","created":{"date-parts":[[2021,6,24]],"date-time":"2021-06-24T23:22:14Z","timestamp":1624576934000},"page":"1125","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Note on Decomposable and Reducible Integer Matrices"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0241-8158","authenticated-orcid":false,"given":"Carlos","family":"Mariju\u00e1n","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada\/IMUVA, Universidad de Valladolid, E-47002 Valladolid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3173-5934","authenticated-orcid":false,"given":"Ignacio","family":"Ojeda","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas\/IMUEX, Universidad de Extremadura, E-06071 Badajoz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4583-0852","authenticated-orcid":false,"given":"Alberto","family":"Vigneron-Tenorio","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas\/INDESS, Universidad de C\u00e1diz, E-11406 Jerez de la Frontera, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.jsc.2013.05.009","article-title":"On decomposable semigroups and applications","volume":"58","year":"2013","journal-title":"J. Symb. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Cox, D., Little, B., and Schenck, H. (2011). Toric Varieties, American Mathematical Society. Graduate Studies in Mathematics.","DOI":"10.1090\/gsm\/124"},{"key":"ref_3","unstructured":"Miller, E., and Sturmfels, B. (2005). Combinatorial Commutative Algebra, Springer. Vol. 227 of Graduate Texts in Mathematics."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Rosales, J.C., and Garc\u00eda-S\u00e1nchez, P.A. (1999). Finitely Generated Commutative Monoids, Nova Science Publishers, Inc.","DOI":"10.1006\/jabr.1999.8039"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1463","DOI":"10.1214\/009053606000000263","article-title":"On the toric algebra of graphical models","volume":"34","author":"Geiger","year":"2006","journal-title":"Ann. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1675","DOI":"10.1016\/j.jnt.2010.01.017","article-title":"Fast computation of Hermite normal forms of random integer matrices","volume":"130","author":"Clement","year":"2010","journal-title":"J. Number Theory"},{"key":"ref_7","unstructured":"The GAP Group (2021, June 23). GAP\u2014Groups, Algorithms, and Programming, Version 4.11.1. Available online: https:\/\/www.gap-system.org."},{"key":"ref_8","unstructured":"(2021). Mathematica, Version 12.3, Wolfram Research, Inc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/0024-3795(94)90486-3","article-title":"Laplacian matrices of graphs: A survey","volume":"197","author":"Merris","year":"1994","journal-title":"Linear Algebra Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1125\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:23:31Z","timestamp":1760163811000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1125"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,24]]},"references-count":9,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071125"],"URL":"https:\/\/doi.org\/10.3390\/sym13071125","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,6,24]]}}}