{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T05:54:20Z","timestamp":1774590860590,"version":"3.50.1"},"reference-count":11,"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":11150,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1983,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Given two (positive) equivalence relations ~<jats:sub>1<\/jats:sub>, ~<jats:sub>2<\/jats:sub> on the set <jats:italic>\u03c9<\/jats:italic> of natural numbers, we say that ~<jats:sub>1<\/jats:sub> is <jats:italic>m-reducible<\/jats:italic> to ~<jats:sub>2<\/jats:sub> if there exists a total recursive function <jats:italic>h<\/jats:italic> such that for every <jats:italic>x, y<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic>, we have <jats:italic>x<\/jats:italic> ~<jats:sub>1<\/jats:sub><jats:italic>y<\/jats:italic> iff <jats:italic>hx<\/jats:italic> ~<jats:sub>2<\/jats:sub><jats:italic>hy<\/jats:italic>. We prove that the equivalence relation induced in <jats:italic>\u03c9<\/jats:italic> by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a \u201cuniformity property\u201d holds). This result allows us to state a classification theorem for positive equivalence relations (Theorem 2). We show that there exist nonisomorphic positive equivalence relations which are complete with respect to the above reducibility; in particular, we discuss the <jats:italic>provable equivalence<\/jats:italic> of a strong enough theory: this relation is complete with respect to reducibility but it does not correspond to a precomplete numeration.<\/jats:p><jats:p>From this fact we deduce that an equivalence relation on <jats:italic>\u03c9<\/jats:italic> can be strongly represented by a formula (see Definition 8) iff it is positive. At last, we interpret the situation from a topological point of view. Among other things, we generalize a result of Visser by showing that the topological space corresponding to a partition in e.i. sets is irreducible and we prove that the set of equivalence classes of true sentences is dense in the Lindenbaum algebra of the theory.<\/jats:p>","DOI":"10.2307\/2273443","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:04:45Z","timestamp":1146938685000},"page":"529-538","source":"Crossref","is-referenced-by-count":62,"title":["Classifying positive equivalence relations"],"prefix":"10.1017","volume":"48","author":[{"given":"Claudio","family":"Bernardi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrea","family":"Sorbi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200037683_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01837553"},{"key":"S0022481200037683_ref002","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19610071108"},{"key":"S0022481200037683_ref008","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200037683_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19750210164"},{"key":"S0022481200037683_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02218645"},{"key":"S0022481200037683_ref010","first-page":"403","article-title":"Numerazioni positive, r.e. classi e formule","volume":"1\u2013B","author":"Sorbi","year":"1982","journal-title":"Unione Matematica Italiana. Bolletino. B"},{"key":"S0022481200037683_ref006","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19550010205"},{"key":"S0022481200037683_ref003","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19730191901"},{"key":"S0022481200037683_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF01976264"},{"key":"S0022481200037683_ref009","doi-asserted-by":"publisher","DOI":"10.1515\/9781400882007-002"},{"key":"S0022481200037683_ref011","first-page":"259","volume-title":"To H.B. Curry: Essays on combinatory logic, lambda calculus and formalism","author":"Visser","year":"1980"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200037683","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T18:37:05Z","timestamp":1558636625000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200037683\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,9]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1983,9]]}},"alternative-id":["S0022481200037683"],"URL":"https:\/\/doi.org\/10.2307\/2273443","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,9]]}}}