{"id":"https://openalex.org/W2281106124","doi":"https://doi.org/10.1007/978-3-319-25754-9_31","title":"Formalizing Lattice-Theoretical Aspects of Rough and Fuzzy Sets","display_name":"Formalizing Lattice-Theoretical Aspects of Rough and Fuzzy Sets","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2281106124","doi":"https://doi.org/10.1007/978-3-319-25754-9_31","mag":"2281106124"},"language":"en","primary_location":{"id":"doi:10.1007/978-3-319-25754-9_31","is_oa":true,"landing_page_url":"https://doi.org/10.1007/978-3-319-25754-9_31","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"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":"book series"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},"type":"book-chapter","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1007/978-3-319-25754-9_31","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5073856205","display_name":"Adam Grabowski","orcid":"https://orcid.org/0000-0001-5026-3990"},"institutions":[{"id":"https://openalex.org/I245364917","display_name":"University of Bia\u0142ystok","ror":"https://ror.org/01qaqcf60","country_code":"PL","type":"education","lineage":["https://openalex.org/I245364917"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Adam Grabowski","raw_affiliation_strings":["Institute of Informatics, University of Bia\u0142ystok, Cio\u0142kowskiego 1M, 15-245, Bia\u0142ystok, Poland"],"affiliations":[{"raw_affiliation_string":"Institute of Informatics, University of Bia\u0142ystok, Cio\u0142kowskiego 1M, 15-245, Bia\u0142ystok, Poland","institution_ids":["https://openalex.org/I245364917"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5085141878","display_name":"Takashi Mitsuishi","orcid":null},"institutions":[{"id":"https://openalex.org/I59596033","display_name":"University of Marketing and Distribution Sciences","ror":"https://ror.org/05k4ves29","country_code":"JP","type":"education","lineage":["https://openalex.org/I59596033"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Takashi Mitsuishi","raw_affiliation_strings":["University of Marketing and Distribution Sciences, Kobe, 3-1 Gakuen Nishimachi Nishi-ku, Kobe, 655-2188, Japan"],"affiliations":[{"raw_affiliation_string":"University of Marketing and Distribution Sciences, Kobe, 3-1 Gakuen Nishimachi Nishi-ku, Kobe, 655-2188, Japan","institution_ids":["https://openalex.org/I59596033"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5073856205"],"corresponding_institution_ids":["https://openalex.org/I245364917"],"apc_list":{"value":5000,"currency":"EUR","value_usd":5392},"apc_paid":{"value":5000,"currency":"EUR","value_usd":5392},"fwci":2.1568,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.87464046,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":null,"issue":null,"first_page":"347","last_page":"356"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11063","display_name":"Rough Sets and Fuzzy Logic","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/T11063","display_name":"Rough Sets and Fuzzy Logic","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/T10215","display_name":"Semantic Web and Ontologies","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"}},{"id":"https://openalex.org/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9886999726295471,"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/rough-set","display_name":"Rough set","score":0.8339035511016846},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.7556169033050537},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.5622016191482544},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.49959611892700195},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.4978010654449463},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.49525466561317444},{"id":"https://openalex.org/keywords/formal-concept-analysis","display_name":"Formal concept analysis","score":0.45153552293777466},{"id":"https://openalex.org/keywords/focus","display_name":"Focus (optics)","score":0.42733651399612427},{"id":"https://openalex.org/keywords/automated-theorem-proving","display_name":"Automated theorem proving","score":0.41447895765304565},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.35806718468666077},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3391339182853699}],"concepts":[{"id":"https://openalex.org/C111012933","wikidata":"https://www.wikidata.org/wiki/Q3137210","display_name":"Rough set","level":2,"score":0.8339035511016846},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.7556169033050537},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.5622016191482544},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.49959611892700195},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.4978010654449463},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.49525466561317444},{"id":"https://openalex.org/C76363472","wikidata":"https://www.wikidata.org/wiki/Q1437394","display_name":"Formal concept analysis","level":2,"score":0.45153552293777466},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.42733651399612427},{"id":"https://openalex.org/C206880738","wikidata":"https://www.wikidata.org/wiki/Q431667","display_name":"Automated theorem proving","level":2,"score":0.41447895765304565},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.35806718468666077},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3391339182853699},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/978-3-319-25754-9_31","is_oa":true,"landing_page_url":"https://doi.org/10.1007/978-3-319-25754-9_31","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"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":"book series"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"}],"best_oa_location":{"id":"doi:10.1007/978-3-319-25754-9_31","is_oa":true,"landing_page_url":"https://doi.org/10.1007/978-3-319-25754-9_31","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"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":"book series"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W139714950","https://openalex.org/W935710377","https://openalex.org/W1494553505","https://openalex.org/W1530333737","https://openalex.org/W1535538569","https://openalex.org/W1556028890","https://openalex.org/W1557923305","https://openalex.org/W1589305208","https://openalex.org/W1596666806","https://openalex.org/W1789208025","https://openalex.org/W1855754411","https://openalex.org/W1975593923","https://openalex.org/W1980605059","https://openalex.org/W1983426508","https://openalex.org/W1987251161","https://openalex.org/W2027654459","https://openalex.org/W2042059286","https://openalex.org/W2054416947","https://openalex.org/W2055266411","https://openalex.org/W2060832326","https://openalex.org/W2069069851","https://openalex.org/W2071050975","https://openalex.org/W2082266564","https://openalex.org/W2093261176","https://openalex.org/W2128085487","https://openalex.org/W2912565176","https://openalex.org/W3100298706","https://openalex.org/W4205529794","https://openalex.org/W4211007335","https://openalex.org/W4232953319"],"related_works":["https://openalex.org/W59961068","https://openalex.org/W1527640492","https://openalex.org/W3041607660","https://openalex.org/W2114938325","https://openalex.org/W2376429569","https://openalex.org/W2978519593","https://openalex.org/W2091110802","https://openalex.org/W2102746356","https://openalex.org/W2018750854","https://openalex.org/W4390066334"],"abstract_inverted_index":{"Fuzzy":[0],"sets":[1,4,42,78],"and":[2,76,148],"rough":[3,56,75],"are":[5,43],"well-known":[6],"approaches":[7],"to":[8,46,79,94],"incomplete":[9],"or":[10,89],"imprecise":[11],"data.":[12],"In":[13],"the":[14,26,31,37,47,52,71,81,103,114,121,129,145],"paper":[15,110],"we":[16],"briefly":[17],"report":[18],"how":[19],"these":[20,152],"frameworks":[21],"were":[22],"successfully":[23],"encoded":[24],"with":[25,113],"help":[27],"of":[28,30,74,83,105,120,124,144,151],"one":[29,119],"leading":[32],"computer":[33],"proof":[34,107],"assistants":[35],"in":[36,102],"world.":[38],"Even":[39],"though":[40],"fuzzy":[41,77],"much":[44],"closer":[45],"set":[48],"theory":[49],"implemented":[50],"within":[51],"Mizar":[53,130],"library":[54],"than":[55],"sets,":[57],"lattices":[58],"as":[59,91,93],"a":[60,64],"basic":[61],"viewpoint":[62],"appeared":[63],"very":[65],"feasible":[66],"one.":[67],"We":[68],"focus":[69],"on":[70,142],"lattice-theoretical":[72],"aspects":[73],"enable":[80],"application":[82],"external":[84],"theorem":[85],"provers":[86],"like":[87],"EQP":[88],"Prover9":[90],"well":[92],"translate":[95],"them":[96],"into":[97],"TPTP":[98],"format":[99],"widely":[100],"recognized":[101],"world":[104],"automated":[106],"search.":[108],"The":[109],"is":[111],"illustrated":[112],"examples":[115],"taken":[116],"just":[117],"from":[118],"largest":[122],"repositories":[123],"computer-checked":[125],"mathematical":[126],"knowledge":[127],"\u2013":[128],"Mathematical":[131],"Library.":[132],"Our":[133],"formal":[134],"development":[135],"allows":[136],"both":[137],"for":[138],"further":[139],"generalizations,":[140],"building":[141],"top":[143],"existing":[146],"knowledge,":[147],"even":[149],"merging":[150],"approaches.":[153]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}