{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,6]],"date-time":"2025-10-06T18:22:41Z","timestamp":1759774961526},"reference-count":18,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,11,13]],"date-time":"2006-11-13T00:00:00Z","timestamp":1163376000000},"content-version":"vor","delay-in-days":5430,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1992,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We will prove in Zermelo\u2010Fraenkel set theory <jats:italic>without axiom of choice<\/jats:italic> that the transitive hull <jats:italic>R<\/jats:italic>* of a relation <jats:italic>R<\/jats:italic> is not much \u201cbigger\u201d than <jats:italic>R<\/jats:italic> itself. As a measure for the size of a relation we introduce the notion of \u03ba<jats:sup>+<\/jats:sup>\u2010<jats:italic>narrowness<\/jats:italic> using <jats:italic>surjective<\/jats:italic> Hartogs numbers rather than the usul <jats:italic>injective<\/jats:italic> Hartogs values. The main theorem of this paper states that the transitive hull of a \u03ba<jats:sup>+<\/jats:sup>\u2010narrow relation is \u03ba<jats:sup>+<\/jats:sup>\u2010narrow. As an immediate corollary we obtain that, for every infinite cardinal \u03ba, the class HC<jats:sub>\u03ba<\/jats:sub> of all \u03ba\u2010hereditary sets is a <jats:italic>set<\/jats:italic> with von Neumann rank \u03f1(HC\u03ba) \u2264 \u03ba<jats:sup>+<\/jats:sup>. Moreover, \u03f1(HC\u03ba) = \u03ba<jats:sup>+<\/jats:sup> if and only if \u03ba is singular, otherwise \u03f1(HC\u03ba) = \u03ba. The statements of the corollary are well known in the presence of the axiom of choice (AC). To prove them without AC \u2010 as carried through here \u2010 is, however, much harder. A special case of the corollary (\u03ba = \u03c9<jats:sub>1<\/jats:sub>, i.e., the class HC\u03c9<jats:sub>1<\/jats:sub> of all hereditarily countable sets) has been treated independently by<jats:sc> T. JECH<\/jats:sc>.<\/jats:p>","DOI":"10.1002\/malq.19920380137","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T05:49:37Z","timestamp":1180417777000},"page":"387-398","source":"Crossref","is-referenced-by-count":3,"title":["ON THE TRANSITIVE HULL OF A \u03ba\u2010NARROW RELATION"],"prefix":"10.1002","volume":"38","author":[{"given":"Karl\u2010Heinz","family":"Diener","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K.\u2010H.","family":"Diener","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,11,13]]},"reference":[{"key":"e_1_2_1_2_2","unstructured":"Diener K.\u2010H. Zur Theorie der absolut freien und verwandter Algebren. Dissertation Universit\u00e4t zu K\u00f6ln1963."},{"key":"e_1_2_1_3_2","doi-asserted-by":"crossref","first-page":"63","DOI":"10.4064\/cm-14-1-63-72","article-title":"Order in absolutely free and related algebras","volume":"14","author":"Diener K.\u2010H.","year":"1966","journal-title":"Colloq. Math."},{"key":"e_1_2_1_4_2","first-page":"357","article-title":"On constructing infinitary languages L\u03b1\u03b2 without the axiom of choice","volume":"29","author":"Diener K.\u2010H.","year":"1983","journal-title":"This Zeitschrift"},{"key":"e_1_2_1_5_2","unstructured":"Diener K.\u2010H. An application of infinitary Universal Algebra to set theory. To appear."},{"key":"e_1_2_1_6_2","unstructured":"Felscher W. Naive Mengen und abstrakte Zahlen I. II. III. Bibliographisches Institut Mannheim\u2010Wien\u2010Z\u00fcrich 1978\/1979."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760939"},{"key":"e_1_2_1_8_2","unstructured":"Gr\u00e4tzer G. Universal Algebra. Princeton1958."},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1002\/mana.19660310104"},{"key":"e_1_2_1_10_2","volume-title":"Cylindric Algebras, Part I.","author":"Henkin L.","year":"1971"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01370734"},{"key":"e_1_2_1_12_2","volume-title":"Model Theory for Infinitary Logic","author":"Keisler H. J.","year":"1971"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02308-2"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.2307\/2273380"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1515\/crll.1952.190.65"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100032898"},{"key":"e_1_2_1_17_2","doi-asserted-by":"crossref","unstructured":"Schmidt J. \u00dcber die Dimension einer partiellen Algebra mit endlichen oder unendlichen Operationen. This Zeitschrift 11 (1965) 227\u2013239.","DOI":"10.1002\/malq.19650110307"},{"key":"e_1_2_1_18_2","article-title":"The theory of abstract algebras with infinitary operations","volume":"18","author":"Slomi\u0144ski J.","year":"1959","journal-title":"Rozprawy Mat."},{"key":"e_1_2_1_19_2","volume-title":"Set Theory","author":"Vaught R. L.","year":"1985"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.19920380137","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.19920380137","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T20:36:21Z","timestamp":1695846981000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.19920380137"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,1]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1992,1]]}},"alternative-id":["10.1002\/malq.19920380137"],"URL":"https:\/\/doi.org\/10.1002\/malq.19920380137","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,1]]}}}