{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T15:09:29Z","timestamp":1648652969522},"reference-count":4,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1927,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2008,12]]},"abstract":"<jats:p>In response to a question of Farah, \u201cHow many Boolean algebras <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004035_inline1\" \/> are there?\u201d [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004035_inline2\" \/> with <jats:italic>I<\/jats:italic> a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:<\/jats:p><jats:p>(i) \u201cBorel ideal\u201d may be improved to \u201canalytic P-ideal\u201d<\/jats:p><jats:p>(ii) \u201ccontinuum-many\u201d may be improved to \u201c<jats:italic>E<\/jats:italic><jats:sub>0<\/jats:sub>-many\u201d; that is, <jats:italic>E<\/jats:italic><jats:sub>0<\/jats:sub> is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.<\/jats:p><jats:p>See [Oli04].<\/jats:p><jats:p>In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set <jats:italic>B<\/jats:italic> a sort of \u201cdirect sum\u201d of the equivalence relations corresponding to the reals in <jats:italic>B<\/jats:italic>. Then if <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>, \u2286 <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub> it was easy to see that the equivalence relation thus induced by <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub> was Borel reducible to the one induced by <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub>, whereas in the opposite case, taking <jats:italic>x<\/jats:italic> to be some element of <jats:italic>B<\/jats:italic> \/ <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub>, it was possible to show that the equivalence relation corresponding to <jats:italic>x<\/jats:italic>, which was part of the equivalence relation induced by <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>, was not Borel reducible to the equivalence relation corresponding to <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub>.<\/jats:p>","DOI":"10.2178\/jsl\/1230396922","type":"journal-article","created":{"date-parts":[[2008,12,27]],"date-time":"2008-12-27T16:55:26Z","timestamp":1230396926000},"page":"1328-1340","source":"Crossref","is-referenced-by-count":0,"title":["Borel complexity of isomorphism between quotient Boolean algebras"],"prefix":"10.1017","volume":"73","author":[{"given":"Su","family":"Gao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael Ray","family":"Oliver","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200004035_ref004","first-page":"799","volume":"69","author":"Oliver","year":"2004","journal-title":"Continuum-many Boolean algebras of the form , I Borel"},{"key":"S0022481200004035_ref003","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S0022481200004035_ref002","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1215\/ijm\/1258138463","article-title":"How many Boolean algebras  are there?","volume":"46","author":"Farah","year":"2002","journal-title":"Illinois Journal of Mathematics"},{"key":"S0022481200004035_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-00-00341-6"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200004035","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T21:36:21Z","timestamp":1556660181000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200004035\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,12]]},"references-count":4,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,12]]}},"alternative-id":["S0022481200004035"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1230396922","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,12]]}}}