{"id":"https://openalex.org/W7134250252","doi":"https://doi.org/10.1016/j.tcs.2026.115879","title":"There is no prime functional digraph: Seifert\u2019s proof revisited","display_name":"There is no prime functional digraph: Seifert\u2019s proof revisited","publication_year":2026,"publication_date":"2026-03-09","ids":{"openalex":"https://openalex.org/W7134250252","doi":"https://doi.org/10.1016/j.tcs.2026.115879"},"language":"en","primary_location":{"id":"doi:10.1016/j.tcs.2026.115879","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.tcs.2026.115879","pdf_url":null,"source":{"id":"https://openalex.org/S90727058","display_name":"Theoretical Computer Science","issn_l":"0304-3975","issn":["0304-3975","1879-2294"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theoretical Computer Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1016/j.tcs.2026.115879","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5082953016","display_name":"Adrien Richard","orcid":"https://orcid.org/0000-0003-3360-5869"},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I4210106479","display_name":"Laboratoire d'Informatique, Signaux et Syst\u00e8mes de Sophia Antipolis","ror":"https://ror.org/01215r597","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I201841394","https://openalex.org/I4210106479","https://openalex.org/I4210159245"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Adrien Richard","raw_affiliation_strings":["Computer Science, CNRS Provence and Corsica Delegation, Laboratoire I3S, 2000 route des Lucioles, Sophia-Antipolis, 06903, France"],"raw_orcid":"https://orcid.org/0000-0003-3360-5869","affiliations":[{"raw_affiliation_string":"Computer Science, CNRS Provence and Corsica Delegation, Laboratoire I3S, 2000 route des Lucioles, Sophia-Antipolis, 06903, France","institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I4210106479"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5082953016"],"corresponding_institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I4210106479"],"apc_list":{"value":2690,"currency":"USD","value_usd":2690},"apc_paid":{"value":2690,"currency":"USD","value_usd":2690},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.54656974,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"1072","issue":null,"first_page":"115879","last_page":"115879"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.22010000050067902,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10849","display_name":"Finite Group Theory Research","score":0.22010000050067902,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11166","display_name":"Analytic Number Theory Research","score":0.11649999767541885,"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"}},{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.08760000020265579,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.47189998626708984},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3287999927997589},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.30070000886917114},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.24719999730587006}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5611000061035156},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.47189998626708984},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4088999927043915},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4081999957561493},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3287999927997589},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3086000084877014},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.30070000886917114},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2922999858856201},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.24719999730587006},{"id":"https://openalex.org/C2318724","wikidata":"https://www.wikidata.org/wiki/Q852732","display_name":"Proof theory","level":3,"score":0.24220000207424164}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1016/j.tcs.2026.115879","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.tcs.2026.115879","pdf_url":null,"source":{"id":"https://openalex.org/S90727058","display_name":"Theoretical Computer Science","issn_l":"0304-3975","issn":["0304-3975","1879-2294"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theoretical Computer Science","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1016/j.tcs.2026.115879","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.tcs.2026.115879","pdf_url":null,"source":{"id":"https://openalex.org/S90727058","display_name":"Theoretical Computer Science","issn_l":"0304-3975","issn":["0304-3975","1879-2294"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theoretical Computer Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"},{"id":"https://openalex.org/F4320334322","display_name":"HORIZON EUROPE Framework Programme","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W194234195","https://openalex.org/W1971224531","https://openalex.org/W1980767716","https://openalex.org/W2046538693","https://openalex.org/W2058221907","https://openalex.org/W2065782501","https://openalex.org/W2076350440","https://openalex.org/W2128084896","https://openalex.org/W2130156088","https://openalex.org/W2902246075","https://openalex.org/W2964274798","https://openalex.org/W4312221151","https://openalex.org/W4393029278","https://openalex.org/W4393183211","https://openalex.org/W4411458742","https://openalex.org/W4414634499","https://openalex.org/W4416505834","https://openalex.org/W4416542316"],"related_works":[],"abstract_inverted_index":{"A":[0,54,68,93,107],"functional":[1,27,47,52,59,91,119,181,221],"digraph":[2,6,48,53,60,71,222],"is":[3,65,73,80,173,206],"a":[4,12,30,46,51,58,154,186,209],"finite":[5],"in":[7,123,136,151,153,160,168],"which":[8],"each":[9,232],"vertex":[10],"has":[11,33],"unique":[13],"out-neighbor.":[14],"Considered":[15],"up":[16],"to":[17,67,75,129,207],"isomorphism":[18],"and":[19,25,87,94,122,176,230],"endowed":[20],"with":[21],"the":[22,82,85,97,171,177,227],"directed":[23],"sum":[24],"product,":[26,86],"digraphs":[28,92,120,182],"form":[29],"semigroup":[31],"that":[32,63,99,104,131,141,161,166,218],"recently":[34],"attracted":[35],"significant":[36],"attention,":[37],"particularly":[38],"regarding":[39],"its":[40],"multiplicative":[41],"structure.":[42],"In":[43,111],"this":[44,142,198,204],"context,":[45],"X":[49,72,100,105],"divides":[50,101,106],"if":[55,78],"there":[56],"exists":[57,223],"Y":[61],"such":[62],"XY":[64],"isomorphic":[66],".":[69,110],"The":[70,158,201],"said":[74],"be":[76],"prime":[77,118,180,220],"it":[79],"not":[81],"identity":[83],"for":[84,89],"if,":[88],"all":[90],"B":[95,109],",":[96],"fact":[98],"AB":[102],"implies":[103],"or":[108],"2020,":[112],"Antonio":[113],"E.":[114],"Porreca":[115],"asked":[116],"whether":[117],"exist,":[121],"2023,":[124],"his":[125],"work":[126,162],"led":[127],"him":[128],"conjecture":[130],"they":[132],"do":[133],"not.":[134],"However,":[135],"2024,":[137],"Barbora":[138],"Hudcov\u00e1":[139],"discovered":[140],"result":[143],"had":[144],"already":[145],"been":[146],"proven":[147],"by":[148,225],"Ralph":[149],"Seifert":[150],"1971,":[152],"somewhat":[155],"forgotten":[156],"paper.":[157],"terminology":[159],"differs":[163],"significantly":[164],"from":[165],"used":[167],"recent":[169],"studies,":[170],"framework":[172],"more":[174,211],"general,":[175],"non-existence":[178],"of":[179,188,203,214],"appears":[183],"only":[184],"as":[185,234,236],"part":[187],"broader":[189],"results,":[190],"relying":[191],"on":[192],"(overly)":[193],"technical":[194],"lemmas":[195],"developed":[196],"within":[197],"general":[199],"setting.":[200],"aim":[202],"note":[205],"present":[208],"much":[210,235],"accessible":[212],"version":[213],"Seifert\u2019s":[215],"proof":[216],"\u2014":[217,224],"no":[219],"using":[226],"current":[228],"language":[229],"simplifying":[231],"step":[233],"possible.":[237]},"counts_by_year":[],"updated_date":"2026-05-05T08:41:31.759640","created_date":"2026-03-10T00:00:00"}