{"id":"https://openalex.org/W2164699613","doi":"https://doi.org/10.2478/v10037-009-0030-x","title":"On the Lattice of Intervals and Rough Sets","display_name":"On the Lattice of Intervals and Rough Sets","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2164699613","doi":"https://doi.org/10.2478/v10037-009-0030-x","mag":"2164699613"},"language":"en","primary_location":{"id":"doi:10.2478/v10037-009-0030-x","is_oa":true,"landing_page_url":"https://doi.org/10.2478/v10037-009-0030-x","pdf_url":"https://sciendo.com/pdf/10.2478/v10037-009-0030-x","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-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","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://sciendo.com/pdf/10.2478/v10037-009-0030-x","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"]},{"id":"https://openalex.org/I4210137370","display_name":"Institute of Mathematics","ror":"https://ror.org/04hrrh248","country_code":"PL","type":"facility","lineage":["https://openalex.org/I4210137370","https://openalex.org/I99542240"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Adam Grabowski","raw_affiliation_strings":["Institute of Mathematics University of Biaystok Akademicka 2, 15-267 Biaystok Poland","Institute of Mathematics University of Bia\u0142ystok Akademicka 2, 15-267 Bia\u0142ystok Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics University of Biaystok Akademicka 2, 15-267 Biaystok Poland","institution_ids":["https://openalex.org/I245364917"]},{"raw_affiliation_string":"Institute of Mathematics University of Bia\u0142ystok Akademicka 2, 15-267 Bia\u0142ystok Poland","institution_ids":["https://openalex.org/I245364917","https://openalex.org/I4210137370"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5042613676","display_name":"Magdalena Jastrz\u0119bska","orcid":"https://orcid.org/0000-0001-8906-6388"},"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"]},{"id":"https://openalex.org/I4210137370","display_name":"Institute of Mathematics","ror":"https://ror.org/04hrrh248","country_code":"PL","type":"facility","lineage":["https://openalex.org/I4210137370","https://openalex.org/I99542240"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Magdalena Jastrz\u0119bska","raw_affiliation_strings":["Institute of Mathematics University of Bia\u0142ystok Akademicka 2, 15-267 Bia\u0142ystok Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics University of Bia\u0142ystok Akademicka 2, 15-267 Bia\u0142ystok Poland","institution_ids":["https://openalex.org/I245364917","https://openalex.org/I4210137370"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5073856205"],"corresponding_institution_ids":["https://openalex.org/I245364917","https://openalex.org/I4210137370"],"apc_list":null,"apc_paid":null,"fwci":0.709,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.73807512,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"17","issue":"4","first_page":"237","last_page":"244"},"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/T11727","display_name":"Advanced Algebra and Logic","score":0.9650999903678894,"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/T10050","display_name":"Multi-Criteria Decision Making","score":0.947700023651123,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/rough-set","display_name":"Rough set","score":0.8917359113693237},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8316134214401245},{"id":"https://openalex.org/keywords/equivalence-relation","display_name":"Equivalence relation","score":0.6838418841362},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.6043781638145447},{"id":"https://openalex.org/keywords/continuation","display_name":"Continuation","score":0.5258023142814636},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.5249790549278259},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.4963567852973938},{"id":"https://openalex.org/keywords/dominance-based-rough-set-approach","display_name":"Dominance-based rough set approach","score":0.4739016592502594},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4191533327102661},{"id":"https://openalex.org/keywords/data-mining","display_name":"Data mining","score":0.1919415295124054},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11162737011909485},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08866769075393677}],"concepts":[{"id":"https://openalex.org/C111012933","wikidata":"https://www.wikidata.org/wiki/Q3137210","display_name":"Rough set","level":2,"score":0.8917359113693237},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8316134214401245},{"id":"https://openalex.org/C156103551","wikidata":"https://www.wikidata.org/wiki/Q130998","display_name":"Equivalence relation","level":2,"score":0.6838418841362},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.6043781638145447},{"id":"https://openalex.org/C88626702","wikidata":"https://www.wikidata.org/wiki/Q1128903","display_name":"Continuation","level":2,"score":0.5258023142814636},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.5249790549278259},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.4963567852973938},{"id":"https://openalex.org/C39105242","wikidata":"https://www.wikidata.org/wiki/Q5290286","display_name":"Dominance-based rough set approach","level":3,"score":0.4739016592502594},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4191533327102661},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.1919415295124054},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11162737011909485},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08866769075393677},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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":11,"locations":[{"id":"doi:10.2478/v10037-009-0030-x","is_oa":true,"landing_page_url":"https://doi.org/10.2478/v10037-009-0030-x","pdf_url":"https://sciendo.com/pdf/10.2478/v10037-009-0030-x","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-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.178.7915","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.178.7915","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://mizar.org/fm/2009-17/pdf17-4/interva1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.579.3440","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.579.3440","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://markun.cs.shinshu-u.ac.jp/mirror/mizar/fm/fm17-4/interva1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.606.7595","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.606.7595","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://fm.uwb.edu.pl/pub/fm/output/9870.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.616.7050","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.616.7050","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://fm.uwb.edu.pl/pub/fm/output/10223.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.618.7422","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.618.7422","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://fm.uwb.edu.pl/pub/fm/output/10200.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.620.2721","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.620.2721","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://fm.uwb.edu.pl/pub/fm/output/10147.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.622.5708","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.622.5708","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://fm.uwb.edu.pl/pub/fm/output/10148.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.630.4054","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.630.4054","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://fm.uwb.edu.pl/pub/fm/output/10158.pdf","raw_type":"text"},{"id":"pmh:oai:doaj.org/article:3bfc597ae508481581c7be10521c8127","is_oa":false,"landing_page_url":"https://doaj.org/article/3bfc597ae508481581c7be10521c8127","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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":"Formalized Mathematics, Vol 17, Iss 4, Pp 237-244 (2009)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/3550","is_oa":false,"landing_page_url":"http://hdl.handle.net/11320/3550","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.2478/v10037-009-0030-x","is_oa":true,"landing_page_url":"https://doi.org/10.2478/v10037-009-0030-x","pdf_url":"https://sciendo.com/pdf/10.2478/v10037-009-0030-x","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-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2164699613.pdf","grobid_xml":"https://content.openalex.org/works/W2164699613.grobid-xml"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W192809544","https://openalex.org/W1483101703","https://openalex.org/W1573317636","https://openalex.org/W1890412369","https://openalex.org/W1986619836","https://openalex.org/W1987699575","https://openalex.org/W2155453192","https://openalex.org/W2163641194","https://openalex.org/W2304533832","https://openalex.org/W4255833381","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W4389318318","https://openalex.org/W2158629008","https://openalex.org/W2167108569","https://openalex.org/W2064989302","https://openalex.org/W2135460823","https://openalex.org/W2600379663","https://openalex.org/W2385807600","https://openalex.org/W1899771913","https://openalex.org/W2220343220","https://openalex.org/W2305735113"],"abstract_inverted_index":{"On":[0],"the":[1,30,35,38,41,46,49,66,69,80,89,102,111,114],"Lattice":[2],"of":[3,22,29,37,71,82,91,104,113,117],"Intervals":[4],"and":[5,40,94],"Rough":[6,8],"Sets":[7],"sets,":[9,74,119],"developed":[10],"by":[11],"Pawlak":[12],"[6],":[13],"are":[14,77,96],"an":[15,83],"important":[16],"tool":[17],"to":[18,79],"describe":[19],"a":[20,58,108],"situation":[21],"incomplete":[23],"or":[24,48,123],"partially":[25],"unknown":[26],"information.":[27],"One":[28],"algebraic":[31,115],"models":[32],"deals":[33],"with":[34,68],"pair":[36,70],"upper":[39],"lower":[42],"approximation.":[43],"Although":[44],"usually":[45],"tolerance":[47],"equivalence":[50],"relation":[51],"is":[52,106],"taken":[53],"into":[54],"account":[55],"when":[56],"considering":[57],"rough":[59,92,118],"set,":[60],"here":[61],"we":[62,76],"rather":[63],"concentrate":[64],"on":[65],"model":[67],"two":[72],"definable":[73],"hence":[75],"close":[78],"notion":[81],"interval":[84],"set.":[85],"In":[86],"this":[87],"article,":[88],"lattices":[90],"sets":[93],"intervals":[95],"formalized.":[97],"This":[98],"paper,":[99],"being":[100],"essentially":[101],"continuation":[103],"[3],":[105],"also":[107],"step":[109],"towards":[110],"formalization":[112],"theory":[116],"as":[120],"in":[121],"[4]":[122],"[9].":[124]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}