{"id":"https://openalex.org/W1983321159","doi":"https://doi.org/10.1002/malq.200410027","title":"Properly ?2 minimal degrees and 0? complementation","display_name":"Properly ?2 minimal degrees and 0? complementation","publication_year":2005,"publication_date":"2005-03-18","ids":{"openalex":"https://openalex.org/W1983321159","doi":"https://doi.org/10.1002/malq.200410027","mag":"1983321159"},"language":"en","primary_location":{"id":"doi:10.1002/malq.200410027","is_oa":false,"landing_page_url":"https://doi.org/10.1002/malq.200410027","pdf_url":null,"source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0044-3050","issn":["0044-3050","0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"MLQ","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076862599","display_name":"S. Barry Cooper","orcid":"https://orcid.org/0000-0001-5103-0400"},"institutions":[{"id":"https://openalex.org/I130828816","display_name":"University of Leeds","ror":"https://ror.org/024mrxd33","country_code":"GB","type":"education","lineage":["https://openalex.org/I130828816"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"S. Barry Cooper","raw_affiliation_strings":["Department of Pure Mathematics, School of Mathematics, University of Leeds, Leeds, LS2 9JT, U. K"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Pure Mathematics, School of Mathematics, University of Leeds, Leeds, LS2 9JT, U. K","institution_ids":["https://openalex.org/I130828816"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111765854","display_name":"Andrew E. M. Lewis","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Andrew E. M. Lewis","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5100624668","display_name":"Yue Yang","orcid":"https://orcid.org/0000-0002-0288-356X"},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":true,"raw_author_name":"Yue Yang","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 119260"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 119260","institution_ids":["https://openalex.org/I165932596"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5076862599","https://openalex.org/A5100624668","https://openalex.org/A5111765854"],"corresponding_institution_ids":["https://openalex.org/I130828816","https://openalex.org/I165932596"],"apc_list":{"value":2630,"currency":"USD","value_usd":2630},"apc_paid":null,"fwci":0.3326,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.59350706,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"51","issue":"3","first_page":"274","last_page":"276"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9998999834060669,"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/T11567","display_name":"semigroups and automata theory","score":0.9904999732971191,"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/T12162","display_name":"Cellular Automata and Applications","score":0.984000027179718,"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/complement","display_name":"Complement (music)","score":0.7612789273262024},{"id":"https://openalex.org/keywords/turing","display_name":"Turing","score":0.7162647247314453},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.6520006060600281},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6481342911720276},{"id":"https://openalex.org/keywords/complementation","display_name":"Complementation","score":0.5960646867752075},{"id":"https://openalex.org/keywords/join","display_name":"Join (topology)","score":0.46580061316490173},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4258509874343872},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34887367486953735},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2642526626586914},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.11627349257469177},{"id":"https://openalex.org/keywords/biology","display_name":"Biology","score":0.07579654455184937},{"id":"https://openalex.org/keywords/genetics","display_name":"Genetics","score":0.05716085433959961},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.048102349042892456}],"concepts":[{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.7612789273262024},{"id":"https://openalex.org/C9870796","wikidata":"https://www.wikidata.org/wiki/Q490481","display_name":"Turing","level":2,"score":0.7162647247314453},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6520006060600281},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6481342911720276},{"id":"https://openalex.org/C188082640","wikidata":"https://www.wikidata.org/wiki/Q1780899","display_name":"Complementation","level":4,"score":0.5960646867752075},{"id":"https://openalex.org/C2776124973","wikidata":"https://www.wikidata.org/wiki/Q3183033","display_name":"Join (topology)","level":2,"score":0.46580061316490173},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4258509874343872},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34887367486953735},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2642526626586914},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.11627349257469177},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.07579654455184937},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.05716085433959961},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.048102349042892456},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C127716648","wikidata":"https://www.wikidata.org/wiki/Q104053","display_name":"Phenotype","level":3,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1002/malq.200410027","is_oa":false,"landing_page_url":"https://doi.org/10.1002/malq.200410027","pdf_url":null,"source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0044-3050","issn":["0044-3050","0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"MLQ","raw_type":"journal-article"},{"id":"pmh:oai:scholarbank.nus.edu.sg:10635/103979","is_oa":false,"landing_page_url":"http://scholarbank.nus.edu.sg/handle/10635/103979","pdf_url":null,"source":{"id":"https://openalex.org/S7407052290","display_name":"National University of Singapore","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Scopus","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":8,"referenced_works":["https://openalex.org/W1528242579","https://openalex.org/W2021757832","https://openalex.org/W2042499838","https://openalex.org/W2073364886","https://openalex.org/W2102988122","https://openalex.org/W2105112555","https://openalex.org/W2167930025","https://openalex.org/W6669471030"],"related_works":["https://openalex.org/W1518010816","https://openalex.org/W1976496938","https://openalex.org/W2189178478","https://openalex.org/W2154338693","https://openalex.org/W1587647201","https://openalex.org/W4205996836","https://openalex.org/W1500684591","https://openalex.org/W3214148052","https://openalex.org/W1988188960","https://openalex.org/W2036810953"],"abstract_inverted_index":{"We":[0],"show":[1],"that":[2,14,27,38],"there":[3],"exists":[4],"a":[5,30,36,41],"properly":[6],"\u03a32":[7],"minimal":[8],"(Turing)":[9],"degree":[10,35],"b,":[11],"and":[12],"moreover":[13],"b":[15,28],"can":[16],"be":[17],"chosen":[18],"to":[19,23],"join":[20],"with":[21],"0\u2032":[22,39],"0\u2033":[24,31],"\u2013":[25],"so":[26],"is":[29],"complement":[32],"for":[33],"every":[34],"such":[37],"\u2264":[40],"<":[42],"0\u2033.":[43],"(\u00a9":[44],"2005":[45],"WILEY-VCH":[46],"Verlag":[47],"GmbH":[48],"&":[49],"Co.":[50],"KGaA,":[51],"Weinheim)":[52]},"counts_by_year":[{"year":2013,"cited_by_count":1}],"updated_date":"2026-06-21T06:12:48.119943","created_date":"2016-06-24T00:00:00"}