{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T00:04:21Z","timestamp":1648598661545},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10968,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,3]]},"abstract":"We say that a pair of r.e. sets B<\/jats:italic> and C<\/jats:italic> split an r.e. set A<\/jats:italic> if B<\/jats:italic> \u2229 C<\/jats:italic> = \u2205 and B<\/jats:italic> \u222a C<\/jats:italic> = A<\/jats:italic>. Friedberg [F] was the first to study the degrees of splittings of r.e. sets. He showed that every nonrecursive r.e. set A has a splitting into nonrecursive sets. Generalizations and strengthenings of Friedberg's result were obtained by Sacks [Sa2], Owings [O], and Morley and Soare [MS].<\/jats:p>The question which motivated both [LR] and this paper is the determination of possible degrees of splittings of A<\/jats:italic>. It is easy to see that if B<\/jats:italic> and C<\/jats:italic> split A<\/jats:italic>, then both B<\/jats:italic> and C<\/jats:italic> are Turing reducible to A<\/jats:italic> (written B<\/jats:italic> \u2264T<\/jats:italic><\/jats:sub>A<\/jats:italic> and C<\/jats:italic> \u2264T<\/jats:italic><\/jats:sub>A<\/jats:italic>). The Sacks splitting theorem [Sa2] is a result in this direction, as are results by Lachlan and Ladner on mitotic and nonmitotic sets. Call an r.e. set A mitotic<\/jats:italic> if there is a splitting B<\/jats:italic> and C<\/jats:italic> of A<\/jats:italic> such that both B<\/jats:italic> and C<\/jats:italic> have the same Turing degree as A<\/jats:italic>; A<\/jats:italic> is nonmitotic<\/jats:italic> otherwise. Lachlan [Lac] showed that nonmitotic sets exist, and Ladner [Lad1], [Lad2] carried out an exhaustive study of the degrees of mitotic sets.<\/jats:p>The Sacks splitting theorem [Sa2] shows that if A<\/jats:italic> is r.e. and nonrecursive, then there are r.e. sets B<\/jats:italic> and C<\/jats:italic> splitting A<\/jats:italic> such that B<\/jats:italic> <T<\/jats:italic><\/jats:sub>A<\/jats:italic> and C<\/jats:italic> <T<\/jats:italic><\/jats:sub>A<\/jats:italic>. Since B<\/jats:italic> is r.e. and nonrecursive, we can now split B<\/jats:italic> and continue in this manner to produce infinitely many r.e. degrees below the degree of A<\/jats:italic> which are degrees of sets forming part of a splitting of A<\/jats:italic>. We say that an r.e. set A<\/jats:italic> has the universal splitting property<\/jats:italic> (USP) if for any r.e. set D<\/jats:italic> \u2264T<\/jats:sub> A<\/jats:italic>, there is a splitting B<\/jats:italic> and C<\/jats:italic> of A<\/jats:italic> such that B<\/jats:italic> and D<\/jats:italic> are Turing equivalent (written B<\/jats:italic> \u2261T<\/jats:italic><\/jats:sub>D<\/jats:italic>).<\/jats:p>","DOI":"10.2307\/2274097","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:07:02Z","timestamp":1146938822000},"page":"137-150","source":"Crossref","is-referenced-by-count":14,"title":["The universal splitting property. II"],"prefix":"10.1017","volume":"49","author":[{"given":"M.","family":"Lerman","sequence":"first","affiliation":[]},{"given":"J. B.","family":"Remmel","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200034009_ref002","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19670130102"},{"key":"S0022481200034009_ref008","doi-asserted-by":"publisher","DOI":"10.2307\/1970393"},{"key":"S0022481200034009_ref001","first-page":"305","volume":"23","author":"Friedberg","year":"1958","journal-title":"Three theorems on recursive enumeration"},{"key":"S0022481200034009_ref003","first-page":"199","volume":"38","author":"Ladner","year":"1973","journal-title":"Mitotic recursively enumerable sets"},{"key":"S0022481200034009_ref005","first-page":"181","volume-title":"Logic Colloquium '80","author":"Lerman","year":"1982"},{"key":"S0022481200034009_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1973-0398805-7"},{"key":"S0022481200034009_ref006","doi-asserted-by":"publisher","DOI":"10.4064\/fm-90-1-45-52"},{"key":"S0022481200034009_ref007","first-page":"173","volume":"32","author":"Owings","year":"1967","journal-title":"Recursion, metarecursion and inclusion"},{"key":"S0022481200034009_ref009","volume-title":"Annals of Mathematical Studies","author":"Sacks","year":"1966"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200034009","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T17:26:22Z","timestamp":1558632382000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200034009\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,3]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1984,3]]}},"alternative-id":["S0022481200034009"],"URL":"http:\/\/dx.doi.org\/10.2307\/2274097","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":["Logic","Philosophy"],"published":{"date-parts":[[1984,3]]}}}