{"id":"https://openalex.org/W1997576179","doi":"https://doi.org/10.2307/2273468","title":"Closure properties of almost-finiteness classes in recursive function theory","display_name":"Closure properties of almost-finiteness classes in recursive function theory","publication_year":1983,"publication_date":"1983-09-01","ids":{"openalex":"https://openalex.org/W1997576179","doi":"https://doi.org/10.2307/2273468","mag":"1997576179"},"language":"en","primary_location":{"id":"doi:10.2307/2273468","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273468","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","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/A5085012955","display_name":"Heinrich Rolletschek","orcid":null},"institutions":[{"id":"https://openalex.org/I86501945","display_name":"University of Delaware","ror":"https://ror.org/01sbq1a82","country_code":"US","type":"education","lineage":["https://openalex.org/I86501945"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Heinrich Rolletschek","raw_affiliation_strings":["University of Delaware, Newark, Delaware 19711","**University of Delaware, Newark, Delaware 19711"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Delaware, Newark, Delaware 19711","institution_ids":["https://openalex.org/I86501945"]},{"raw_affiliation_string":"**University of Delaware, Newark, Delaware 19711","institution_ids":["https://openalex.org/I86501945"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5085012955"],"corresponding_institution_ids":["https://openalex.org/I86501945"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.12179279,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"48","issue":"3","first_page":"756","last_page":"763"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9297000169754028,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9297000169754028,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/cartesian-product","display_name":"Cartesian product","score":0.9031531810760498},{"id":"https://openalex.org/keywords/join","display_name":"Join (topology)","score":0.8231068849563599},{"id":"https://openalex.org/keywords/closure","display_name":"Closure (psychology)","score":0.8187662363052368},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7328163385391235},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6052887439727783},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5689428448677063},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.5226978063583374},{"id":"https://openalex.org/keywords/closed-set","display_name":"Closed set","score":0.49002230167388916},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.46640127897262573},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4636470079421997},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4384463429450989},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4339643716812134},{"id":"https://openalex.org/keywords/natural-number","display_name":"Natural number","score":0.4317159056663513},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1392715871334076},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.06672811508178711}],"concepts":[{"id":"https://openalex.org/C65236422","wikidata":"https://www.wikidata.org/wiki/Q173740","display_name":"Cartesian product","level":2,"score":0.9031531810760498},{"id":"https://openalex.org/C2776124973","wikidata":"https://www.wikidata.org/wiki/Q3183033","display_name":"Join (topology)","level":2,"score":0.8231068849563599},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.8187662363052368},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7328163385391235},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6052887439727783},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5689428448677063},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.5226978063583374},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.49002230167388916},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.46640127897262573},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4636470079421997},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4384463429450989},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4339643716812134},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.4317159056663513},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1392715871334076},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.06672811508178711},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","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}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2273468","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273468","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jsl/1183741335","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183741335","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W2000487993","https://openalex.org/W2097898753","https://openalex.org/W2117832390","https://openalex.org/W2152079305","https://openalex.org/W2530006810","https://openalex.org/W2905135108"],"related_works":["https://openalex.org/W3214148052","https://openalex.org/W2151692181","https://openalex.org/W4248476017","https://openalex.org/W2102840537","https://openalex.org/W4205189031","https://openalex.org/W2112422394","https://openalex.org/W2368412539","https://openalex.org/W2993274418","https://openalex.org/W2557103432","https://openalex.org/W2998146897"],"abstract_inverted_index":{"In":[0,282],"this":[1,52],"paper":[2],"we":[3,176,284],"will":[4,50,67],"investigate":[5],"closure":[6,60,75],"properties":[7,61,76],"of":[8,11,15,20,22,87,89,289,311],"the":[9,74,81,179,287,309,314,323,326],"class":[10,86,288,310],"immune":[12],"sets":[13,21,292],"and":[14,27,66,105,136,154,293],"other,":[16],"more":[17],"restricted":[18],"classes":[19,175,294],"natural":[23],"numbers":[24],"under":[25,95,112,126,143,161],"union":[26],"similar":[28],"operations.":[29],"Our":[30,318],"main":[31],"theorem":[32],"states":[33],"that":[34,42],"there":[35],"exists":[36],"a":[37],"hyperhyperimmune":[38],"set":[39],"A":[40,43,45,98,103,115,117,120,129,134,147,152,165,167,170],"such":[41,307],"join":[44,99,116],"is":[46,93,110,124,141,159,187,201,208,216,224,233,241,249,257,265,320],"not":[47],"hyperhyperimmune.":[48],"We":[49,72],"prove":[51],"result":[53],"in":[54,70,80,273,279],"\u00a73.":[55],"The":[56,173,268],"remaining":[57],"questions":[58,315],"about":[59],"are":[62,78,178,270],"all":[63,271],"much":[64],"easier":[65],"be":[68],"answered":[69],"\u00a72.":[71],"consider":[73,177,286],"which":[77],"given":[79],"following":[82],"Definition.":[83],"Let":[84],"bea.":[85],"subsets":[88],"N":[90],".":[91,108,122,139,157,172],"(i)":[92],"closed":[94,111,125,142,160],"join,":[96],"if":[97,114,128,146,164],"B":[100,106,131,137,149,155],"\u2208":[101,104,107,118,121,132,135,138,150,153,156,168,171],"whenever":[102,119,133,151,169],"(ii)":[109],"self-join,":[113],"(iii)":[123],"union,":[127],"\u22c3":[130],"(iv)":[140],"cartesian":[144,162],"product,":[145],"\u00d7":[148,166],"(v)":[158],"self-product,":[163],"almost-finiteness":[174],"following:":[180],"1":[181],"=":[182,190,196,204,211,220,228,236,245,252,261],"{":[183,191,197,205,212,221,229,237,246,253,262],"X":[184,186,192,198,200,206,213,215,222,230,232,238,240,247,254,256,263],"\u2223":[185,193,199,207,214,223,231,239,248,255,264],"immune};":[188,259],"2":[189],"simple};":[194],"3":[195],"hyperimmune};":[202,218],"4":[203],"hypersimple};":[209,226],"5":[210],"strongly":[217,225,242],"6":[219],"7":[227],"hyperhyperimmune};":[234,243],"8":[235],"9":[244],"hyperhypersimple};":[250],"10":[251],"dense":[258,266],"11":[260],"simple}.":[267],"definitions":[269],"contained":[272],"Rogers":[274,299],"[4,":[275,300],"Chapter":[276],"12]":[277],"and/or":[278],"Robinson":[280],"[3].":[281],"addition":[283],"could":[285],"-strongly":[290],"hyperimmune":[291],"defined":[295],"by":[296],"lim-properties;":[297],"see":[298],"pp.":[301],"243\u2013244].":[302],"For":[303],"other":[304],"classes,":[305],"however,":[306],"as":[308],"cohesive":[312],"sets,":[313],"become":[316],"trivial.":[317],"aim":[319],"to":[321],"establish":[322],"table":[324],"on":[325],"next":[327],"page.":[328]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}