{"id":"https://openalex.org/W935710377","doi":"https://doi.org/10.2478/forma-2014-0032","title":"The Formal Construction of Fuzzy Numbers","display_name":"The Formal Construction of Fuzzy Numbers","publication_year":2014,"publication_date":"2014-12-01","ids":{"openalex":"https://openalex.org/W935710377","doi":"https://doi.org/10.2478/forma-2014-0032","mag":"935710377"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2014-0032","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2014-0032","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2014-0032","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.sciendo.com/pdf/10.2478/forma-2014-0032","any_repository_has_fulltext":true},"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 Akademicka 2, 15-267 Bia\u0142ystok Poland"],"affiliations":[{"raw_affiliation_string":"Institute of Informatics University of Bia\u0142ystok Akademicka 2, 15-267 Bia\u0142ystok Poland","institution_ids":["https://openalex.org/I245364917"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5073856205"],"corresponding_institution_ids":["https://openalex.org/I245364917"],"apc_list":null,"apc_paid":null,"fwci":1.8403,"has_fulltext":true,"cited_by_count":9,"citation_normalized_percentile":{"value":0.84302113,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"22","issue":"4","first_page":"321","last_page":"327"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11063","display_name":"Rough Sets and Fuzzy Logic","score":0.9940000176429749,"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":0.9940000176429749,"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/T10820","display_name":"Fuzzy Logic and Control Systems","score":0.972100019454956,"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/T11106","display_name":"Data Management and Algorithms","score":0.9336000084877014,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/mathematics","display_name":"Mathematics","score":0.6865675449371338},{"id":"https://openalex.org/keywords/type-2-fuzzy-sets-and-systems","display_name":"Type-2 fuzzy sets and systems","score":0.6581008434295654},{"id":"https://openalex.org/keywords/fuzzy-number","display_name":"Fuzzy number","score":0.635809600353241},{"id":"https://openalex.org/keywords/fuzzy-set-operations","display_name":"Fuzzy set operations","score":0.6156453490257263},{"id":"https://openalex.org/keywords/fuzzy-classification","display_name":"Fuzzy classification","score":0.5933683514595032},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.5739174485206604},{"id":"https://openalex.org/keywords/membership-function","display_name":"Membership function","score":0.5719041228294373},{"id":"https://openalex.org/keywords/fuzzy-mathematics","display_name":"Fuzzy mathematics","score":0.5504686236381531},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.546983003616333},{"id":"https://openalex.org/keywords/defuzzification","display_name":"Defuzzification","score":0.4643927216529846},{"id":"https://openalex.org/keywords/usable","display_name":"USable","score":0.4540936350822449},{"id":"https://openalex.org/keywords/fuzzy-measure-theory","display_name":"Fuzzy measure theory","score":0.44517478346824646},{"id":"https://openalex.org/keywords/base","display_name":"Base (topology)","score":0.4262150228023529},{"id":"https://openalex.org/keywords/generality","display_name":"Generality","score":0.42485353350639343},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40563473105430603},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.38083136081695557},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.3624500632286072},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.31300681829452515},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.30160588026046753},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.17388316988945007}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6865675449371338},{"id":"https://openalex.org/C17350324","wikidata":"https://www.wikidata.org/wiki/Q7860605","display_name":"Type-2 fuzzy sets and systems","level":5,"score":0.6581008434295654},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.635809600353241},{"id":"https://openalex.org/C148671577","wikidata":"https://www.wikidata.org/wiki/Q5511133","display_name":"Fuzzy set operations","level":4,"score":0.6156453490257263},{"id":"https://openalex.org/C127385683","wikidata":"https://www.wikidata.org/wiki/Q1475696","display_name":"Fuzzy classification","level":4,"score":0.5933683514595032},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.5739174485206604},{"id":"https://openalex.org/C5263885","wikidata":"https://www.wikidata.org/wiki/Q1967302","display_name":"Membership function","level":4,"score":0.5719041228294373},{"id":"https://openalex.org/C166586474","wikidata":"https://www.wikidata.org/wiki/Q5511129","display_name":"Fuzzy mathematics","level":5,"score":0.5504686236381531},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.546983003616333},{"id":"https://openalex.org/C170260401","wikidata":"https://www.wikidata.org/wiki/Q5251735","display_name":"Defuzzification","level":5,"score":0.4643927216529846},{"id":"https://openalex.org/C2780615836","wikidata":"https://www.wikidata.org/wiki/Q2471869","display_name":"USable","level":2,"score":0.4540936350822449},{"id":"https://openalex.org/C100516043","wikidata":"https://www.wikidata.org/wiki/Q4455138","display_name":"Fuzzy measure theory","level":5,"score":0.44517478346824646},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.4262150228023529},{"id":"https://openalex.org/C2780767217","wikidata":"https://www.wikidata.org/wiki/Q5532421","display_name":"Generality","level":2,"score":0.42485353350639343},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40563473105430603},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.38083136081695557},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3624500632286072},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.31300681829452515},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.30160588026046753},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.17388316988945007},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0},{"id":"https://openalex.org/C542102704","wikidata":"https://www.wikidata.org/wiki/Q183257","display_name":"Psychotherapist","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C136764020","wikidata":"https://www.wikidata.org/wiki/Q466","display_name":"World Wide Web","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2478/forma-2014-0032","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2014-0032","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2014-0032","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:86b3afda-d592-4c28-8e64-0afed7479f02/oai:repozytorium.uwb.edu.pl:11320/3728","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/3728","pdf_url":null,"source":{"id":"https://openalex.org/S4306401248","display_name":"Jagiellonian University Repository (Jagiellonian University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I126596746","host_organization_name":"Jagiellonian University","host_organization_lineage":["https://openalex.org/I126596746"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""},{"id":"pmh:oai:doaj.org/article:a72f80e203a24c708180e3d19130dbf7","is_oa":true,"landing_page_url":"https://doaj.org/article/a72f80e203a24c708180e3d19130dbf7","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Formalized Mathematics, Vol 22, Iss 4, Pp 321-327 (2014)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.2478/forma-2014-0032","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2014-0032","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2014-0032","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.4099999964237213}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W935710377.pdf","grobid_xml":"https://content.openalex.org/works/W935710377.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W180239714","https://openalex.org/W192809544","https://openalex.org/W260340778","https://openalex.org/W1596666806","https://openalex.org/W1938825438","https://openalex.org/W2027654459","https://openalex.org/W2069069851","https://openalex.org/W2096867974","https://openalex.org/W2143851127","https://openalex.org/W2163641194","https://openalex.org/W2296232473","https://openalex.org/W2314125776","https://openalex.org/W2330022088","https://openalex.org/W2554327433","https://openalex.org/W2912565176","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2674573885","https://openalex.org/W2037428035","https://openalex.org/W2166637398","https://openalex.org/W3186561448","https://openalex.org/W2375182298","https://openalex.org/W2048452567","https://openalex.org/W2347394627","https://openalex.org/W2025902904","https://openalex.org/W2573957424","https://openalex.org/W22698454"],"abstract_inverted_index":{"Summary":[0],"In":[1],"this":[2,148],"article,":[3],"we":[4,52,67,100,146,171],"continue":[5],"the":[6,9,19,24,33,37,60,79,89,102,112,120,140,159,167],"development":[7],"of":[8,11,26,36,59,81,104,114,122,139,158,164,166],"theory":[10],"fuzzy":[12,27,71,109,116,124,128,175],"sets":[13,72,110,129],"[23],":[14],"started":[15],"with":[16,18,185],"[14]":[17],"future":[20,179],"aim":[21,172],"to":[22,45,136],"provide":[23],"formalization":[25,198],"numbers":[28,176],"[8]":[29],"in":[30,43,50,64,144,177,191],"terms":[31],"reflecting":[32],"current":[34],"state":[35],"Mizar":[38],"Mathematical":[39],"Library.":[40],"Note":[41],"that":[42,54],"order":[44],"have":[46],"more":[47],"usable":[48],"approach":[49,188],"[14],":[51],"revised":[53],"article":[55],"as":[56,74,111,181,183,189],"well;":[57],"some":[58],"ideas":[61],"were":[62,130],"described":[63],"[12].":[65],"As":[66],"can":[68],"actually":[69],"understand":[70],"just":[73],"their":[75,85],"membership":[76,82],"functions":[77],"(via":[78],"equality":[80],"function":[83],"and":[84,107,126,143,193,201],"set-theoretic":[86],"counterpart),":[87],"all":[88],"calculations":[90],"are":[91],"much":[92],"simpler.":[93],"To":[94],"test":[95],"our":[96,178],"newly":[97],"proposed":[98],"approach,":[99,168],"give":[101],"notions":[103],"(normal)":[105],"triangular":[106],"trapezoidal":[108],"examples":[113],"concrete":[115],"objects.":[117],"Also":[118],"-cuts,":[119],"core":[121],"a":[123],"set,":[125],"normalized":[127],"defined.":[131],"Main":[132],"technical":[133],"obstacle":[134],"was":[135,199],"prove":[137],"continuity":[138],"glued":[141],"maps,":[142],"fact":[145],"did":[147],"not":[149],"through":[150],"its":[151],"topological":[152],"counterpart,":[153],"but":[154],"extensively":[155],"reusing":[156],"properties":[157],"real":[160],"line":[161],"(with":[162],"loss":[163],"generality":[165],"though),":[169],"because":[170],"at":[173],"formalizing":[174],"submissions,":[180],"well":[182],"merging":[184],"rough":[186],"set":[187],"introduced":[190],"[13]":[192],"[11].":[194],"Our":[195],"base":[196],"for":[197],"[9]":[200],"[10].":[202]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2015,"cited_by_count":2}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2016-06-24T00:00:00"}