{"id":"https://openalex.org/W2045473234","doi":"https://doi.org/10.1007/bf01786987","title":"A note on complete sets and transitive closure","display_name":"A note on complete sets and transitive closure","publication_year":1981,"publication_date":"1981-12-01","ids":{"openalex":"https://openalex.org/W2045473234","doi":"https://doi.org/10.1007/bf01786987","mag":"2045473234"},"language":"en","primary_location":{"id":"doi:10.1007/bf01786987","is_oa":false,"landing_page_url":"https://doi.org/10.1007/bf01786987","pdf_url":null,"source":{"id":"https://openalex.org/S4210177266","display_name":"Theory of Computing Systems","issn_l":"1432-4350","issn":["1432-4350","1433-0490"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Systems Theory","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/A5103477007","display_name":"Ronald V. Book","orcid":null},"institutions":[{"id":"https://openalex.org/I154570441","display_name":"University of California, Santa Barbara","ror":"https://ror.org/02t274463","country_code":"US","type":"education","lineage":["https://openalex.org/I154570441"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ronald V. Book","raw_affiliation_strings":["Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California, USA","Department of Mathematics, University of California at Santa Barbara,Santa Barbara,USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California, USA","institution_ids":["https://openalex.org/I154570441"]},{"raw_affiliation_string":"Department of Mathematics, University of California at Santa Barbara,Santa Barbara,USA","institution_ids":["https://openalex.org/I154570441"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5049670210","display_name":"Celia Wrathall","orcid":null},"institutions":[{"id":"https://openalex.org/I154570441","display_name":"University of California, Santa Barbara","ror":"https://ror.org/02t274463","country_code":"US","type":"education","lineage":["https://openalex.org/I154570441"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Celia Wrathall","raw_affiliation_strings":["Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California, USA","Department of Mathematics, University of California at Santa Barbara,Santa Barbara,USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California, USA","institution_ids":["https://openalex.org/I154570441"]},{"raw_affiliation_string":"Department of Mathematics, University of California at Santa Barbara,Santa Barbara,USA","institution_ids":["https://openalex.org/I154570441"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.21898138,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"15","issue":"1","first_page":"311","last_page":"313"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11596","display_name":"Constraint Satisfaction and Optimization","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11596","display_name":"Constraint Satisfaction and Optimization","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11727","display_name":"Advanced Algebra and Logic","score":0.9966999888420105,"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/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9914000034332275,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/transitive-closure","display_name":"Transitive closure","score":0.960677981376648},{"id":"https://openalex.org/keywords/transitive-relation","display_name":"Transitive relation","score":0.8753896355628967},{"id":"https://openalex.org/keywords/closure","display_name":"Closure (psychology)","score":0.8190298676490784},{"id":"https://openalex.org/keywords/transitive-reduction","display_name":"Transitive reduction","score":0.7393817901611328},{"id":"https://openalex.org/keywords/relation","display_name":"Relation (database)","score":0.6466702818870544},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6084652543067932},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.48509061336517334},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.37550854682922363},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.27458781003952026},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.07743826508522034},{"id":"https://openalex.org/keywords/data-mining","display_name":"Data mining","score":0.05557340383529663},{"id":"https://openalex.org/keywords/political-science","display_name":"Political science","score":0.04934856295585632},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.04039359092712402}],"concepts":[{"id":"https://openalex.org/C128896722","wikidata":"https://www.wikidata.org/wiki/Q1501387","display_name":"Transitive closure","level":2,"score":0.960677981376648},{"id":"https://openalex.org/C191399111","wikidata":"https://www.wikidata.org/wiki/Q64861","display_name":"Transitive relation","level":2,"score":0.8753896355628967},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.8190298676490784},{"id":"https://openalex.org/C129691609","wikidata":"https://www.wikidata.org/wiki/Q3088151","display_name":"Transitive reduction","level":5,"score":0.7393817901611328},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.6466702818870544},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6084652543067932},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.48509061336517334},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37550854682922363},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.27458781003952026},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.07743826508522034},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.05557340383529663},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.04934856295585632},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.04039359092712402},{"id":"https://openalex.org/C22149727","wikidata":"https://www.wikidata.org/wiki/Q7940747","display_name":"Voltage graph","level":4,"score":0.0},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/bf01786987","is_oa":false,"landing_page_url":"https://doi.org/10.1007/bf01786987","pdf_url":null,"source":{"id":"https://openalex.org/S4210177266","display_name":"Theory of Computing Systems","issn_l":"1432-4350","issn":["1432-4350","1433-0490"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Systems Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.6399999856948853}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W1987540368","https://openalex.org/W1996184580","https://openalex.org/W2036265926","https://openalex.org/W2065251436","https://openalex.org/W2066356146","https://openalex.org/W2067540335","https://openalex.org/W2069222055"],"related_works":["https://openalex.org/W1553139688","https://openalex.org/W2362009423","https://openalex.org/W2371151875","https://openalex.org/W2383282217","https://openalex.org/W2136009070","https://openalex.org/W1994877243","https://openalex.org/W4230967508","https://openalex.org/W2012063573","https://openalex.org/W1996983051","https://openalex.org/W2349651113"],"abstract_inverted_index":null,"counts_by_year":[{"year":2014,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}