{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,19]],"date-time":"2023-10-19T21:01:19Z","timestamp":1697749279808},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13068,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub> be a fixed, fully effective, infinite dimensional vector space. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049604_inline1\" \/> be the lattice consisting of the recursively enumerable (r.e.) subspaces of <jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub>, under the operations of intersection and weak sum (see \u00a71 for precise definitions). In this article we examine the algebraic properties of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049604_inline1\" \/>.<\/jats:p><jats:p>Early research on recursively enumerable algebraic structures was done by Rabin [14], Fr\u00f6lich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the more recent work concerning vector spaces of Metakides and Nerode [12], Crossley and Nerode [2], Remmel [15], [16], and Kalantari [8].<\/jats:p><jats:p>In the main theorem below, we extend a result of Lachlan from the lattice <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049604_inline2\" \/> of r.e. sets to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049604_inline1\" \/>. We define hyperhypersimple vector spaces, discuss some of their properties and show if <jats:italic>A, B<\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049604_inline1\" \/>, and <jats:italic>A<\/jats:italic> is a hyperhypersimple subspace of <jats:italic>B<\/jats:italic> then there is a recursive space <jats:italic>C<\/jats:italic> such that <jats:italic>A<\/jats:italic> + <jats:italic>C<\/jats:italic> = <jats:italic>B<\/jats:italic>. It will be proven that if <jats:italic>V<\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049604_inline1\" \/> and the lattice of superspaces of <jats:italic>V<\/jats:italic> is a complemented modular lattice then <jats:italic>V<\/jats:italic> is hyperhypersimple. The final section contains a summary of related results concerning maximality and simplicity.<\/jats:p>","DOI":"10.2307\/2272824","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:47:35Z","timestamp":1146952055000},"page":"260-269","source":"Crossref","is-referenced-by-count":9,"title":["Simple and hyperhypersimple vector spaces"],"prefix":"10.1017","volume":"43","author":[{"given":"Allen","family":"Retzlaff","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049604_ref022","doi-asserted-by":"publisher","DOI":"10.2307\/1970842"},{"key":"S0022481200049604_ref021","volume-title":"Annals of Mathematical Studies","author":"Sacks","year":"1966"},{"key":"S0022481200049604_ref019","volume-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik","author":"Retzlaff"},{"key":"S0022481200049604_ref018","unstructured":"Retzlaff A. , Recursive and simple vector spaces, Ph.D. dissertation, Cornell University, 1976."},{"key":"S0022481200049604_ref017","author":"Remmel","journal-title":"On vector spaces with no extendible bases"},{"key":"S0022481200049604_ref016","unstructured":"Remmel J. , Co-recursively enumerable structures, Ph.D. dissertation, Cornell University, 1974."},{"key":"S0022481200049604_ref015","first-page":"341","article-title":"Computable algebra, general theory and theory of computable fields","volume":"95","author":"Rabin","year":"1960","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200049604_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-85933-5"},{"key":"S0022481200049604_ref009","unstructured":"Kalantari I. , Structural properties of the lattice of recursively enumerable vector spaces, Ph.D. dissertation, Cornell University, 1976."},{"key":"S0022481200049604_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1968-0227009-1"},{"key":"S0022481200049604_ref005","first-page":"309","volume":"23","author":"Friedberg","year":"1958","journal-title":"Three theorems on recursive enumeration"},{"key":"S0022481200049604_ref014","doi-asserted-by":"publisher","DOI":"10.4064\/fm-90-1-45-52"},{"key":"S0022481200049604_ref020","volume-title":"Theory of recursive functions and effective computabitity","author":"Rogers","year":"1968"},{"key":"S0022481200049604_ref004","first-page":"363","volume":"34","author":"Dekker","year":"1969","journal-title":"Countable vector spaces with recursive operations, I, II"},{"key":"S0022481200049604_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90015-8"},{"key":"S0022481200049604_ref012","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/BFb0062858","volume-title":"Algebra and logic, Lecture Notes in Mathematics","author":"Metakides","year":"1975"},{"key":"S0022481200049604_ref008","first-page":"85","volume":"35","author":"Hamilton","year":"1970","journal-title":"Bases and \u03b1-dimensions of countable vector spaces with recursive operations"},{"key":"S0022481200049604_ref001","volume-title":"American Mathematical Society Colloquium Publications","volume":"25","author":"Birkhoff","year":"1967"},{"key":"S0022481200049604_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(76)90189-7"},{"key":"S0022481200049604_ref006","first-page":"407","article-title":"Effective procedures in field theory","volume":"284","author":"Fr\u00f6lich","year":"1955","journal-title":"Philosophical Transactions of the Royal Society of London"},{"key":"S0022481200049604_ref007","unstructured":"Guhl R. , Two types of recursively enumerable vector spaces, Ph.D. dissertation, Rutgers University, 1973."},{"key":"S0022481200049604_ref010","first-page":"481","volume":"42","author":"Kalantari","year":"1977","journal-title":"Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049604","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:58:13Z","timestamp":1558987093000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049604\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,6]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1978,6]]}},"alternative-id":["S0022481200049604"],"URL":"https:\/\/doi.org\/10.2307\/2272824","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,6]]}}}