{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T05:09:15Z","timestamp":1772514555153,"version":"3.50.1"},"reference-count":7,"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":8685,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,6]]},"abstract":"<jats:p>An important program in the study of the structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/> the lattice of r.e. sets modulo finite sets, is the classification of the orbits under <jats:italic>Aut<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>), the automorphism group of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>, If \u03a6 \u2208 <jats:italic>Aut<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>) and \u03a6(<jats:italic>W<\/jats:italic><jats:sub>e<\/jats:sub>) =* <jats:italic>W<jats:sub>h(e)<\/jats:sub><\/jats:italic> for all <jats:italic>e<\/jats:italic> \u2208 \u03c9, then <jats:italic>h<\/jats:italic> is called a <jats:italic>presentation<\/jats:italic> of \u03a6. Define <jats:italic>Aut<\/jats:italic><jats:sub>\u03c7<\/jats:sub>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>) to be the class of those elements of <jats:italic>Aut<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>) that have a presentation in the class of functions <jats:italic>X<\/jats:italic>, where for instance <jats:italic>X<\/jats:italic> might be the class of \u0394<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> functions for <jats:italic>n<\/jats:italic> \u2208 \u03c9. If \u03a6 \u2208 <jats:italic>Aut<jats:sub>x<\/jats:sub><\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>) then we will say that \u03a6 is an <jats:italic>X-automorphism<\/jats:italic>. Note that we only need to consider \u0394<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>- and \u03a0<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-orbits and automorphisms since if <jats:italic>f<\/jats:italic> is a \u03a3<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-presentation, then <jats:italic>f<\/jats:italic> is a total function and therefore \u0394<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>.<\/jats:p><jats:p>D<jats:sc>efinition<\/jats:sc>. If <jats:italic>X<\/jats:italic> is a class of functions and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline2\"\/> \u2286 {<jats:italic>W<jats:sub>i<\/jats:sub><\/jats:italic>: <jats:italic>i<\/jats:italic> &lt; \u03c9}, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline2\"\/> is an <jats:italic>X-orbit<\/jats:italic> iff <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline2\"\/> is an orbit under <jats:italic>Aut<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>) and for all <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=\"S0022481200026128_inline2\"\/> there is a \u03a6 \u2208 <jats:italic>Aut<jats:sub>x<\/jats:sub><\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>) satisfying \u03a6(<jats:italic>A<\/jats:italic>) =* <jats:italic>B<\/jats:italic>. (Here \u03a6: <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/> \u2192 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>.)<\/jats:p><jats:p>Harrington proved that the creative sets form a \u0394<jats:sub>0<\/jats:sub>-orbit and Soare proved that the maximal sets form \u0394<jats:sub>3<\/jats:sub>-orbit. We will show that there is no Boolean algebra <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline3\"\/> such that {<jats:italic>A<\/jats:italic>: <jats:italic>A<\/jats:italic> is r.e. and <jats:italic>\u2112<\/jats:italic>*(<jats:italic>A<\/jats:italic>) \u2248 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline3\"\/>] forms a \u0394<jats:sub>2<\/jats:sub>-orbit, where <jats:italic>\u2112<\/jats:italic>*(<jats:italic>A<\/jats:italic>) is the principal filter of <jats:italic>A<\/jats:italic> in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>; <jats:italic>\u2112<\/jats:italic>*(<jats:italic>A<\/jats:italic>) = {<jats:italic>B<\/jats:italic>: <jats:italic>B<\/jats:italic> \u2286* <jats:italic>A<\/jats:italic> &amp; <jats:italic>B<\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026128_inline1\"\/>}. The idea behind the proof of this theorem is very similar to the proof by Soare that maximal sets do not form a \u0394<jats:sub>2<\/jats:sub>-orbit (see Soare [1974] or Soare [1987]).<\/jats:p>","DOI":"10.2307\/2274662","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:35:29Z","timestamp":1146954929000},"page":"744-760","source":"Crossref","is-referenced-by-count":1,"title":["Boolean algebras and orbits of the lattice of r.e sets modulo the finite sets"],"prefix":"10.1017","volume":"55","author":[{"given":"Peter","family":"Cholak","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026128_ref007","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-65-03247-3"},{"key":"S0022481200026128_ref004","first-page":"51","volume":"49","year":"1984","journal-title":"Orbits of hyperhypersimple sets"},{"key":"S0022481200026128_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761377"},{"key":"S0022481200026128_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1968-0227009-1"},{"key":"S0022481200026128_ref001","first-page":"309","volume":"23","author":"Friedberg","year":"1958","journal-title":"Three theorems on recursive enumeration: I: Decomposition. II: Maximal set. III: Enumeration without duplication"},{"key":"S0022481200026128_ref005","doi-asserted-by":"publisher","DOI":"10.2307\/1970842"},{"key":"S0022481200026128_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026128","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T21:14:08Z","timestamp":1558214048000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026128\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1990,6]]}},"alternative-id":["S0022481200026128"],"URL":"https:\/\/doi.org\/10.2307\/2274662","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,6]]}}}