{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T15:53:50Z","timestamp":1698335630268},"reference-count":4,"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":"In this paper we will investigate closure properties of the class of immune sets and of other, more restricted classes of sets of natural numbers under union and similar operations. Our main theorem states that there exists a hyperhyperimmune set A<\/jats:italic> such that A<\/jats:italic> join A<\/jats:italic> is not hyperhyperimmune. We will prove this result in \u00a73. The remaining questions about closure properties are all much easier and will be answered in \u00a72.<\/jats:p>We consider the closure properties which are given in the following<\/jats:p>Definition. Let bea. class of subsets of N<\/jats:italic>.<\/jats:p>(i) is closed under join, if A<\/jats:italic> join B<\/jats:italic> \u2208 whenever A<\/jats:italic> \u2208 and B<\/jats:italic> \u2208 .<\/jats:p>(ii) is closed under self-join, if A<\/jats:italic> join A<\/jats:italic> \u2208 whenever A<\/jats:italic> \u2208 .<\/jats:p>(iii) is closed under union, if A<\/jats:italic> \u22c3 B<\/jats:italic> \u2208 whenever A<\/jats:italic> \u2208 and B<\/jats:italic> \u2208 .<\/jats:p>(iv) is closed under cartesian product, if A<\/jats:italic> \u00d7 B<\/jats:italic> \u2208 whenever A<\/jats:italic> \u2208 and B<\/jats:italic> \u2208 .<\/jats:p>(v) is closed under cartesian self-product, if A<\/jats:italic> \u00d7 A<\/jats:italic> \u2208 whenever A<\/jats:italic> \u2208 .<\/jats:p>The almost-finiteness classes we consider are the following:<\/jats:p>1<\/jats:sub> = {X<\/jats:italic>\u2223X<\/jats:italic> is immune};<\/jats:p>2<\/jats:sub> = {X<\/jats:italic>\u2223 simple};<\/jats:p>3<\/jats:sub> = {X<\/jats:italic>\u2223X<\/jats:italic> is hyperimmune};<\/jats:p>4<\/jats:sub> = {X<\/jats:italic>\u2223 is hypersimple};<\/jats:p>5<\/jats:sub> = {X<\/jats:italic>\u2223X<\/jats:italic> is strongly hyperimmune};<\/jats:p>6<\/jats:sub> = {X<\/jats:italic>\u2223 is strongly hypersimple};<\/jats:p>7<\/jats:sub> = {X<\/jats:italic>\u2223X<\/jats:italic> is hyperhyperimmune};<\/jats:p>8<\/jats:sub> = {X<\/jats:italic>\u2223X<\/jats:italic> is strongly hyperhyperimmune};<\/jats:p>9<\/jats:sub> = {X<\/jats:italic>\u2223 is hyperhypersimple};<\/jats:p>10<\/jats:sub> = {X<\/jats:italic>\u2223X<\/jats:italic> is dense immune};<\/jats:p>11<\/jats:sub> = {X<\/jats:italic>\u2223 is dense simple}.<\/jats:p>The definitions are all contained in Rogers [4, Chapter 12] and\/or in Robinson [3]. In addition we could consider the class of -strongly hyperimmune sets and classes defined by lim-properties; see Rogers [4, pp. 243\u2013244]. For other classes, however, such as the class of cohesive sets, the questions become trivial.<\/jats:p>Our aim is to establish the table on the next page.<\/jats:p>","DOI":"10.2307\/2273468","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:04:45Z","timestamp":1146953085000},"page":"756-763","source":"Crossref","is-referenced-by-count":1,"title":["Closure properties of almost-finiteness classes in recursive function theory"],"prefix":"10.1017","volume":"48","author":[{"given":"Heinrich","family":"Rolletschek","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200037932_ref004","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200037932_ref003","first-page":"162","volume":"32","author":"Robinson","year":"1967","journal-title":"Simplicity of recursively enumerable sets"},{"key":"S0022481200037932_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1953-0058533-7"},{"key":"S0022481200037932_ref001","first-page":"598","volume":"37","author":"Cooper","year":"1972","journal-title":"Jump-equivalence of -hyperhyperimmune sets"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200037932","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T22:37:06Z","timestamp":1558651026000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200037932\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,9]]},"references-count":4,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1983,9]]}},"alternative-id":["S0022481200037932"],"URL":"https:\/\/doi.org\/10.2307\/2273468","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,9]]}}}