{"id":"https://openalex.org/W3193958913","doi":"https://doi.org/10.3390/axioms10030198","title":"Closure System and Its Semantics","display_name":"Closure System and Its Semantics","publication_year":2021,"publication_date":"2021-08-23","ids":{"openalex":"https://openalex.org/W3193958913","doi":"https://doi.org/10.3390/axioms10030198","mag":"3193958913"},"language":"en","primary_location":{"id":"doi:10.3390/axioms10030198","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms10030198","pdf_url":"https://www.mdpi.com/2075-1680/10/3/198/pdf?version=1630722959","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/10/3/198/pdf?version=1630722959","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5039897657","display_name":"Yinbin Lei","orcid":null},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan\u2013Ann Arbor","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yinbin Lei","raw_affiliation_strings":["Department of Psychology, The University of Michigan, Ann Arbor, MI 48109, USA"],"affiliations":[{"raw_affiliation_string":"Department of Psychology, The University of Michigan, Ann Arbor, MI 48109, USA","institution_ids":["https://openalex.org/I27837315"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100433169","display_name":"Jun Zhang","orcid":"https://orcid.org/0000-0002-5532-872X"},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan\u2013Ann Arbor","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Jun Zhang","raw_affiliation_strings":["Department of Psychology, The University of Michigan, Ann Arbor, MI 48109, USA"],"affiliations":[{"raw_affiliation_string":"Department of Psychology, The University of Michigan, Ann Arbor, MI 48109, USA","institution_ids":["https://openalex.org/I27837315"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5100433169"],"corresponding_institution_ids":["https://openalex.org/I27837315"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.148,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.51519806,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"10","issue":"3","first_page":"198","last_page":"198"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11063","display_name":"Rough Sets and Fuzzy Logic","score":0.9986000061035156,"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":0.9986000061035156,"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":0.9979000091552734,"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/T11671","display_name":"Fuzzy and Soft Set Theory","score":0.9919000267982483,"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/closure-operator","display_name":"Closure operator","score":0.7340599298477173},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.7071736454963684},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6539355516433716},{"id":"https://openalex.org/keywords/closure","display_name":"Closure (psychology)","score":0.6146011352539062},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5872877240180969},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.5187433362007141},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.48802676796913147},{"id":"https://openalex.org/keywords/closed-set","display_name":"Closed set","score":0.48009732365608215},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4451168179512024},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.408527672290802},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.23991844058036804},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.19651958346366882},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.18499431014060974},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.11564218997955322}],"concepts":[{"id":"https://openalex.org/C160771489","wikidata":"https://www.wikidata.org/wiki/Q10564851","display_name":"Closure operator","level":3,"score":0.7340599298477173},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.7071736454963684},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6539355516433716},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.6146011352539062},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5872877240180969},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.5187433362007141},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.48802676796913147},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.48009732365608215},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4451168179512024},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.408527672290802},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23991844058036804},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.19651958346366882},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.18499431014060974},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11564218997955322},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","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/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms10030198","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms10030198","pdf_url":"https://www.mdpi.com/2075-1680/10/3/198/pdf?version=1630722959","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:7ca39af764e84f5392f2e3d8795976f2","is_oa":true,"landing_page_url":"https://doaj.org/article/7ca39af764e84f5392f2e3d8795976f2","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 10, Iss 3, p 198 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/10/3/198/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms10030198","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms; Volume 10; Issue 3; Pages: 198","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms10030198","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms10030198","pdf_url":"https://www.mdpi.com/2075-1680/10/3/198/pdf?version=1630722959","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3193958913.pdf","grobid_xml":"https://content.openalex.org/works/W3193958913.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W959633836","https://openalex.org/W1505150927","https://openalex.org/W1993205599","https://openalex.org/W1994170308","https://openalex.org/W1995262757","https://openalex.org/W2057856707","https://openalex.org/W2062238175","https://openalex.org/W2093397547","https://openalex.org/W2141175249","https://openalex.org/W2148443116","https://openalex.org/W2260217400","https://openalex.org/W2313098266","https://openalex.org/W2324116309","https://openalex.org/W2330980058","https://openalex.org/W2396903734","https://openalex.org/W2605555461","https://openalex.org/W2972048725","https://openalex.org/W4233384360","https://openalex.org/W4234269419","https://openalex.org/W4234326533","https://openalex.org/W4240412856","https://openalex.org/W4247424885","https://openalex.org/W4255318870","https://openalex.org/W6630190730","https://openalex.org/W6702381498","https://openalex.org/W6736510629"],"related_works":["https://openalex.org/W2970346870","https://openalex.org/W2380317631","https://openalex.org/W50402501","https://openalex.org/W2034094897","https://openalex.org/W1985909370","https://openalex.org/W12608245","https://openalex.org/W118416546","https://openalex.org/W3145990545","https://openalex.org/W4205189031","https://openalex.org/W2993274418"],"abstract_inverted_index":{"It":[0,49],"is":[1,50],"well":[2],"known":[3,52],"that":[4,53,95,114,128],"topological":[5,12,55,61,107,123,195],"spaces":[6],"are":[7,249],"axiomatically":[8],"characterized":[9],"by":[10,25,93,169],"the":[11,16,84,96,119,129,171,198,241],"closure":[13,56,69,188],"operator":[14,44,57,82],"satisfying":[15],"Kuratowski":[17],"Closure":[18],"Axioms.":[19],"Equivalently,":[20],"they":[21],"can":[22,64,101,160],"be":[23,65,103,161],"axiomatized":[24],"other":[26],"set":[27,99],"operators":[28,100],"encoding":[29],"primitive":[30],"semantics":[31,130],"of":[32,131,177,200,210,223,243,246],"topology,":[33],"such":[34,113],"as":[35,121,174,251],"interior":[36,62,133,180],"operator,":[37,39,41,75,77],"exterior":[38,76,136],"boundary":[40,74,139],"or":[42,217],"derived-set":[43,79],"(or":[45],"dually,":[46,59],"co-derived-set":[47],"operator).":[48],"also":[51],"a":[54,60,138,144,150,157,175,178,186],"(and":[58,80,182],"operator)":[63],"weakened":[66,85,104,232],"into":[67],"generalized":[68,179,187],"(interior)":[70],"systems.":[71,124],"What":[72],"about":[73],"and":[78,219,228,234,245],"co-derived-set)":[81],"in":[83,109,122,208],"systems?":[86],"Our":[87],"paper":[88],"completely":[89],"answers":[90],"this":[91],"question":[92],"showing":[94],"above":[97],"six":[98],"all":[102],"(from":[105],"their":[106,115],"counterparts)":[108],"an":[110,132,135,141,147,164,205,252],"appropriate":[111],"way":[112],"inter-relationships":[116],"remain":[117],"essentially":[118],"same":[120],"Moreover,":[125],"we":[126],"show":[127],"point,":[134,137,140,143,146,149,152],"accumulation":[142],"co-accumulation":[145],"isolated":[148],"repelling":[151],"etc.":[153],"with":[154,202,236],"respect":[155,203],"to":[156,163,193,204,230],"given":[158],"set,":[159,207],"extended":[162],"arbitrary":[165,206],"subset":[166,172],"system":[167,173,181],"simply":[168],"treating":[170],"base":[176],"hence":[183,235],"its":[184,212,220],"dual,":[185],"system).":[189],"This":[190],"allows":[191],"us":[192],"extend":[194],"semantics,":[196],"namely":[197],"characterization":[199],"points":[201],"terms":[209],"both":[211],"spatial":[213],"relations":[214],"(interior,":[215],"exterior,":[216],"boundary)":[218],"dynamic":[221],"convergence":[222],"any":[224],"sequence":[225],"(accumulation,":[226],"co-accumulation,":[227],"isolation),":[229],"much":[231],"systems":[233],"wider":[237],"applicability.":[238],"Examples":[239],"from":[240],"theory":[242],"matroid":[244],"Knowledge/Learning":[247],"Spaces":[248],"used":[250],"illustration.":[253]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}