{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T11:24:21Z","timestamp":1758281061305},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12976,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,9]]},"abstract":"<jats:p>In [4], Metakides and Nerode define a <jats:italic>recursively presented vector space V<jats:sub>\u221e<\/jats:sub><\/jats:italic>. over a (finite or infinite) recursive field <jats:italic>F<\/jats:italic> to consist of a recursive subset <jats:italic>U<\/jats:italic> of the natural numbers <jats:italic>N<\/jats:italic> and operations of vector addition and scalar multiplication which are partial recursive and under which <jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub> becomes a vector space. Throughout this paper, we will identify <jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub> with <jats:italic>N<\/jats:italic>, say via some fixed G\u00f6del numbering, and assume <jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub> is infinite dimensional and has a <jats:italic>dependence algorithm<\/jats:italic>, i.e., there is a uniform effective procedure which determines whether any given <jats:italic>n<\/jats:italic>-tuple <jats:italic>v<\/jats:italic><jats:sub>0<\/jats:sub>, \u2026, <jats:italic>v<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub> from <jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub> is linearly dependent. Given a subspace <jats:italic>W<\/jats:italic> of <jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub>, we write dim(<jats:italic>W<\/jats:italic>) for the dimension of <jats:italic>W<\/jats:italic>. Given subspaces <jats:italic>V<\/jats:italic> and <jats:italic>W<\/jats:italic> of <jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub>, <jats:italic>V + W<\/jats:italic> will denote the weak sum of <jats:italic>V<\/jats:italic> and <jats:italic>W<\/jats:italic> and if <jats:italic>V<\/jats:italic> \u2229 <jats:italic>W<\/jats:italic> = {<jats:bold>0<\/jats:bold>) (where <jats:bold>0<\/jats:bold> is the zero vector of <jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub>), we write <jats:italic>V<\/jats:italic> \u2295 <jats:italic>W<\/jats:italic> instead of <jats:italic>V + W<\/jats:italic>. If <jats:italic>W \u2287 V<\/jats:italic>, we write <jats:italic>W<\/jats:italic> mod <jats:italic>V<\/jats:italic> for the quotient space. An independent set <jats:italic>A<\/jats:italic> \u2286 <jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub> is <jats:italic>extendible<\/jats:italic> if there is a r.e. independent set <jats:italic>I<\/jats:italic> \u2287 <jats:italic>A<\/jats:italic> such that <jats:italic>I \u2212 A<\/jats:italic> is infinite and <jats:italic>A<\/jats:italic> is <jats:italic>nonextendible<\/jats:italic> if it is not the case that <jats:italic>A<\/jats:italic> is extendible.<\/jats:p>","DOI":"10.2307\/2273519","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:48:21Z","timestamp":1146937701000},"page":"430-441","source":"Crossref","is-referenced-by-count":9,"title":["A <i>r<\/i>-maximal vector space not contained in any maximal vector space"],"prefix":"10.1017","volume":"43","author":[{"given":"J.","family":"Remmel","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049318_ref002","first-page":"481","volume":"42","author":"Kalantari","year":"1977","journal-title":"Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces"},{"key":"S0022481200049318_ref005","first-page":"400","volume":"42","author":"Remmel","year":"1977","journal-title":"Maximal and cohesive vector spaces"},{"key":"S0022481200049318_ref001","first-page":"293","volume":"43","author":"Kalantari","year":"1978","journal-title":"Major subspaces of recursively enumerable vector spaces"},{"key":"S0022481200049318_ref007","unstructured":"Retzlaff A. , Recursive and simple vector spaces, Ph.D. thesis, Cornell University, 1976."},{"key":"S0022481200049318_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1968-0227009-1"},{"key":"S0022481200049318_ref008","first-page":"162","volume":"32","author":"Robinson","year":"1967","journal-title":"Simplicity of recursively enumerable sets"},{"key":"S0022481200049318_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90015-8"},{"key":"S0022481200049318_ref006","unstructured":"Remmel J. B. , On r.e. and co-r.e. vector spaces with nonextendible bases, this Journal (to appear)."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049318","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T15:33:36Z","timestamp":1558971216000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049318\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,9]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1978,9]]}},"alternative-id":["S0022481200049318"],"URL":"https:\/\/doi.org\/10.2307\/2273519","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,9]]}}}