{"id":"https://openalex.org/W2013910858","doi":"https://doi.org/10.2307/2273519","title":"A <i>r</i>-maximal vector space not contained in any maximal vector space","display_name":"A <i>r</i>-maximal vector space not contained in any maximal vector space","publication_year":1978,"publication_date":"1978-09-01","ids":{"openalex":"https://openalex.org/W2013910858","doi":"https://doi.org/10.2307/2273519","mag":"2013910858"},"language":"en","primary_location":{"id":"doi:10.2307/2273519","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273519","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/A5113511227","display_name":"Jeffrey B. Remmel","orcid":null},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California, San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"J. Remmel","raw_affiliation_strings":["University of California at San Diego, La Jolla, CA 92093"],"affiliations":[{"raw_affiliation_string":"University of California at San Diego, La Jolla, CA 92093","institution_ids":["https://openalex.org/I36258959"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5113511227"],"corresponding_institution_ids":["https://openalex.org/I36258959"],"apc_list":null,"apc_paid":null,"fwci":1.8981,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.85205596,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"43","issue":"3","first_page":"430","last_page":"441"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9994000196456909,"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":0.9994000196456909,"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"}},{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9993000030517578,"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9943000078201294,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/linear-subspace","display_name":"Linear subspace","score":0.7931597828865051},{"id":"https://openalex.org/keywords/vector-space","display_name":"Vector space","score":0.699414849281311},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6532211303710938},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6243501305580139},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5442720651626587},{"id":"https://openalex.org/keywords/normed-vector-space","display_name":"Normed vector space","score":0.5268662571907043},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.5266050100326538},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.5207743048667908},{"id":"https://openalex.org/keywords/quotient-space","display_name":"Quotient space (topology)","score":0.5007998943328857},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4744904041290283},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.4592987596988678},{"id":"https://openalex.org/keywords/quotient","display_name":"Quotient","score":0.3984754681587219},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.17751047015190125},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.16654786467552185},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07118842005729675},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06898719072341919}],"concepts":[{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.7931597828865051},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.699414849281311},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6532211303710938},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6243501305580139},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5442720651626587},{"id":"https://openalex.org/C171739935","wikidata":"https://www.wikidata.org/wiki/Q726210","display_name":"Normed vector space","level":2,"score":0.5268662571907043},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5266050100326538},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.5207743048667908},{"id":"https://openalex.org/C40753290","wikidata":"https://www.wikidata.org/wiki/Q1139111","display_name":"Quotient space (topology)","level":3,"score":0.5007998943328857},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4744904041290283},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.4592987596988678},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.3984754681587219},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.17751047015190125},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.16654786467552185},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07118842005729675},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06898719072341919},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2273519","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273519","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W1975466840","https://openalex.org/W2062135743","https://openalex.org/W2097898753","https://openalex.org/W2136958030","https://openalex.org/W2145433104","https://openalex.org/W2168321497"],"related_works":["https://openalex.org/W2481226472","https://openalex.org/W2889939100","https://openalex.org/W4246811655","https://openalex.org/W2042334973","https://openalex.org/W1975179576","https://openalex.org/W4320035820","https://openalex.org/W2391321755","https://openalex.org/W2738827407","https://openalex.org/W2127530926","https://openalex.org/W2617085105"],"abstract_inverted_index":{"In":[0],"[4],":[1],"Metakides":[2],"and":[3,34,39,46,72,79,134,150,152,222],"Nerode":[4],"define":[5],"a":[6,15,25,52,81,88,113,208],"recursively":[7],"presented":[8],"vector":[9,37,53,166],"space":[10],"V":[11,49,61,74,107,117,133,137,140,149,154,168,173,178,185,191,201],"\u221e":[12,50,62,75,108,118,138,169,202],".":[13,130,181],"over":[14],"(finite":[16],"or":[17],"infinite)":[18],"recursive":[19,26,45],"field":[20],"F":[21],"to":[22],"consist":[23],"of":[24,29,36,116,128,136,148,167,177],"subset":[27],"U":[28],"the":[30,126,145,164,193,230],"natural":[31],"numbers":[32],"N":[33,64],"operations":[35],"addition":[38],"scalar":[40],"multiplication":[41],"which":[42,48,92],"are":[43],"partial":[44],"under":[47],"becomes":[51],"space.":[54,195],"Throughout":[55],"this":[56],"paper,":[57],"we":[58,120,171,187],"will":[59,143],"identify":[60],"with":[63],",":[65,84,101,119,139,186],"say":[66],"via":[67],"some":[68],"fixed":[69],"G\u00f6del":[70],"numbering,":[71],"assume":[73],"is":[76,87,109,163,203,207,220,224,228,234],"infinite":[77,221],"dimensional":[78],"has":[80],"dependence":[82],"algorithm":[83],"i.e.,":[85],"there":[86,206],"uniform":[89],"effective":[90],"procedure":[91],"determines":[93],"whether":[94],"any":[95],"given":[96],"n":[97,104],"-tuple":[98],"v":[99,103],"0":[100,159,162],"\u2026,":[102],"\u22121":[105],"from":[106],"linearly":[110],"dependent.":[111],"Given":[112,131],"subspace":[114],"W":[115,123,129,135,142,151,156,175,180,183,189],"write":[121,172,188],"dim(":[122],")":[124,160],"for":[125,192],"dimension":[127],"subspaces":[132],"+":[141,179],"denote":[144],"weak":[146],"sum":[147],"if":[153,205,226],"\u2229":[155],"=":[157],"{":[158],"(where":[161],"zero":[165],"),":[170],"\u2295":[174],"instead":[176],"If":[182],"\u2287":[184,213],"mod":[190],"quotient":[194],"An":[196],"independent":[197,210],"set":[198,211],"A":[199,214,219,223,233],"\u2286":[200],"extendible":[204],"r.e.":[209],"I":[212,217],"such":[215],"that":[216,232],"\u2212":[218],"nonextendible":[225],"it":[227],"not":[229],"case":[231],"extendible.":[235]},"counts_by_year":[{"year":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}