{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T06:35:03Z","timestamp":1649054103272},"reference-count":7,"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":11699,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,3]]},"abstract":"<jats:p>Let \u03b5 stand for the set of nonnegative integers (<jats:italic>numbers<\/jats:italic>), <jats:italic>V<\/jats:italic> for the class of all subcollections of <jats:italic>\u03b5<\/jats:italic>(<jats:italic>sets<\/jats:italic>), <jats:italic>\u039b<\/jats:italic> for the set of isols, <jats:italic>\u039b<\/jats:italic><jats:sub><jats:italic>R<\/jats:italic><\/jats:sub> for the set of regressive isols, and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044674_inline01\" \/> for the set of mappings from a subset of <jats:italic>\u03b5<\/jats:italic> into <jats:italic>\u03b5<\/jats:italic> (<jats:italic>functions<\/jats:italic>). If <jats:italic>\u0192<\/jats:italic> is a function we write <jats:italic>\u03b4\u0192<\/jats:italic> and <jats:italic>\u03c1\u0192<\/jats:italic> for its domain and range, respectively. We denote the inclusion relation by \u2283 and proper inclusion by \u228a. The sets <jats:italic>\u03b1<\/jats:italic> and <jats:italic>\u03b2<\/jats:italic> are recursively equivalent [written: <jats:italic>\u03b1 \u2243 \u03b2<\/jats:italic>], if <jats:italic>\u03b4\u0192<\/jats:italic> = <jats:italic>\u03b1<\/jats:italic> and <jats:italic>\u03c1\u0192<\/jats:italic> = <jats:italic>\u03b2<\/jats:italic> for some function <jats:italic>\u0192<\/jats:italic> with a one-to-one partial recursive extension. We denote the recursive equivalence type of a set <jats:italic>\u03b1<\/jats:italic>, {<jats:italic>\u03c3<\/jats:italic> \u2208 <jats:italic>V<\/jats:italic> \u2223 <jats:italic>\u03c3<\/jats:italic> \u2243 <jats:italic>\u03b1<\/jats:italic>} by Req(<jats:italic>\u03b1<\/jats:italic>). The reader is assumed to be familiar with the contents of [1], [2], [3], and [6].<\/jats:p><jats:p>The concept of an <jats:italic>\u03c9<\/jats:italic>-group was introduced in [6], and that of an <jats:italic>\u03c9<\/jats:italic>-homomorphism in [1]. However, except for a few examples, very little is known about the structure of <jats:italic>\u03c9<\/jats:italic>-groups. If <jats:italic>G<\/jats:italic> is an <jats:italic>\u03c9<\/jats:italic>-group and <jats:italic>\u03a0<\/jats:italic> is an <jats:italic>\u03c9<\/jats:italic>-homomorphism, then it follows that <jats:italic>K<\/jats:italic> = Ker <jats:italic>\u03a0<\/jats:italic> and <jats:italic>H<\/jats:italic> = <jats:italic>\u03a0(G)<\/jats:italic> are <jats:italic>\u03c9<\/jats:italic>-groups. The question arises that if we know the structure of <jats:italic>K<\/jats:italic> and <jats:italic>H<\/jats:italic>, then what can we say about the structure of <jats:italic>G<\/jats:italic>? In this paper we will begin the study of <jats:italic>\u03c9<\/jats:italic>-extensions, which will give us a partial answer to this question.<\/jats:p>","DOI":"10.2307\/2273378","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:58:32Z","timestamp":1146938312000},"page":"27-36","source":"Crossref","is-referenced-by-count":1,"title":["An introduction to <i>\u03c9<\/i>-extensions of <i>\u03c9<\/i>-groups"],"prefix":"10.1017","volume":"47","author":[{"given":"C. H.","family":"Applebaum","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044674_ref006","first-page":"13","volume":"34","author":"Hassett","year":"1969","journal-title":"Recursive equivalence types and groups"},{"key":"S0022481200044674_ref007","volume-title":"The theory of groups, an introduction","author":"Rotman","year":"1973"},{"key":"S0022481200044674_ref005","unstructured":"Hassett M. J. , Some theorems on regressive isols and isolic groups, Doctoral Thesis, Rutgers University, 1966."},{"key":"S0022481200044674_ref004","first-page":"67","article-title":"Recursive equivalence types","volume":"3","author":"Dekker","year":"1960","journal-title":"University of California Publications in Mathematics (N.S.)"},{"key":"S0022481200044674_ref002","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093894224"},{"key":"S0022481200044674_ref003","first-page":"559","volume":"35","author":"Applebaum","year":"1970","journal-title":"Partial recursive functions and \u03c9-functions"},{"key":"S0022481200044674_ref001","first-page":"55","volume":"36","author":"Applebaum","year":"1971","journal-title":"\u03c9-homomorphisms and \u03c9-groups"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044674","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T17:42:48Z","timestamp":1558719768000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044674\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1982,3]]}},"alternative-id":["S0022481200044674"],"URL":"https:\/\/doi.org\/10.2307\/2273378","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,3]]}}}