{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T05:36:59Z","timestamp":1776231419952,"version":"3.50.1"},"reference-count":10,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Found. Comput. Sci."],"published-print":{"date-parts":[[2016,2]]},"abstract":"<jats:p> We present a ring theoretic approach to \u010cern\u00fd's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those in \u010cern\u00fd's series. Semisimplicity gives also the advantage of \u201cfactorizing\u201d the problem of finding a synchronizing word into the sub-problems of finding \u201cshort\u201d words that are zeros into the projection of the simple components in the Wedderburn-Artin decomposition. In the general case this last problem is related to the search of radical words of length at most [Formula: see text] where n is the number of states of the automaton. We show that the solution of this \u201cRadical Conjecture\u201d would give an upper bound [Formula: see text] for the shortest reset word in a strongly connected synchronizing automaton. Finally, we use this approach to prove the Radical Conjecture in some particular cases and \u010cern\u00fd's conjecture for the class of strongly semisimple synchronizing automata. These are automata whose sets of synchronizing words are cyclic ideals, or equivalently, ideal regular languages that are closed under taking roots. <\/jats:p>","DOI":"10.1142\/s0129054116400037","type":"journal-article","created":{"date-parts":[[2016,5,4]],"date-time":"2016-05-04T04:25:46Z","timestamp":1462335946000},"page":"127-145","source":"Crossref","is-referenced-by-count":7,"title":["Semisimple Synchronizing Automata and the Wedderburn-Artin Theory"],"prefix":"10.1142","volume":"27","author":[{"given":"Jorge","family":"Almeida","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica, Faculdade de Ci\u00eancias, Universidade do Porto, 4169-007 Porto, Portugal"},{"name":"Rua do Campo Alegre, 687 Porto, 4169-007, Portugal"}]},{"given":"Emanuele","family":"Rodaro","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica, Faculdade de Ci\u00eancias, Universidade do Porto, 4169-007 Porto, Portugal"},{"name":"Rua do Campo Alegre, 687 Porto, 4169-007, Portugal"}]}],"member":"219","published-online":{"date-parts":[[2016,5,4]]},"reference":[{"key":"p_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-08-04712-0"},{"key":"p_5","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2006.02.003"},{"issue":"1","key":"p_6","first-page":"155","volume":"18","author":"Babcs\u00e1nyi I.","year":"2007","journal-title":"Acta Cybernet."},{"key":"p_8","first-page":"208","volume":"14","author":"\u010cern\u00fd J.","year":"1964","journal-title":"Slovensk. Akad. Vied."},{"key":"p_12","doi-asserted-by":"publisher","DOI":"10.1142\/S0129054104002406"},{"key":"p_13","doi-asserted-by":"publisher","DOI":"10.1016\/j.ic.2010.11.020"},{"key":"p_16","first-page":"10","volume":"1","author":"Rystsov I. K.","year":"1997","journal-title":"Issues"},{"key":"p_17","doi-asserted-by":"publisher","DOI":"10.1007\/s10559-015-9742-9"},{"key":"p_19","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2011.06.012"},{"key":"p_22","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2009.03.021"}],"container-title":["International Journal of Foundations of Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0129054116400037","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T15:02:37Z","timestamp":1565103757000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0129054116400037"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2]]},"references-count":10,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2016,5,4]]},"published-print":{"date-parts":[[2016,2]]}},"alternative-id":["10.1142\/S0129054116400037"],"URL":"https:\/\/doi.org\/10.1142\/s0129054116400037","relation":{},"ISSN":["0129-0541","1793-6373"],"issn-type":[{"value":"0129-0541","type":"print"},{"value":"1793-6373","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,2]]}}}