{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:28:33Z","timestamp":1759336113170},"reference-count":12,"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":11334,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1983,3]]},"abstract":"<jats:p>Definition 1 (Kurepa [3, p. 99]). Let <jats:italic>E<\/jats:italic> be a partially ordered set. Then <jats:italic>\u03c3E<\/jats:italic> denotes the set of all bounded well-ordered subsets of <jats:italic>E<\/jats:italic>. We consider <jats:italic>\u03c3E<\/jats:italic> as a partially ordered set with ordering <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200043383_inline1\" \/> defined as follows: <jats:italic>s<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200043383_inline1\" \/><jats:italic>t<\/jats:italic> if and only if <jats:italic>s<\/jats:italic> is an initial segment of <jats:italic>t<\/jats:italic>.<\/jats:p><jats:p>Then <jats:italic>\u03c3E<\/jats:italic> is a tree, i.e., {<jats:italic>s<\/jats:italic> \u2208 <jats:italic>\u03c3 E<\/jats:italic>\u2223 <jats:italic>s<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200043383_inline1\" \/><jats:italic>t<\/jats:italic>} is well-ordered for every <jats:italic>t<\/jats:italic> \u2208 <jats:italic>\u03c3E<\/jats:italic>. The trees of the form <jats:italic>\u03b1E<\/jats:italic> were extensively studied by Kurepa in [3]\u2013[10]. For example, in [4], he used <jats:italic>\u03c3Q<\/jats:italic> and <jats:italic>\u03c3R<\/jats:italic> to construct various sorts of Aronszajn trees. (Here <jats:italic>Q<\/jats:italic> and <jats:italic>R<\/jats:italic> denote the rationals and reals, respectively.) While considering monotone mapping between some kind of ordered sets, he came to the following two questions several times:<\/jats:p><jats:p>P.1. Does there exist a strictly increasing rational function on <jats:italic>\u03c3Q<\/jats:italic>? (See [4, Probl\u00e8me 2], [5, p. 1033], [6, p. 841], [7, Problem 23.3.3].)<\/jats:p><jats:p>P.2. Let <jats:italic>T<\/jats:italic> be a tree in which every chain is countable and every level has cardinality &lt;2<jats:sup>\u2135<jats:sub>0<\/jats:sub><\/jats:sup>. Does there exist a strictly increasing real function on <jats:italic>T<\/jats:italic>? (See [6, p. 246] and [7].)<\/jats:p><jats:p>It is known today that Problem 2 is independent of the usual axioms of set theory (see [1]). Concerning Problem 1 we have the following.<\/jats:p>","DOI":"10.2307\/2273324","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:02:24Z","timestamp":1146952944000},"page":"91-96","source":"Crossref","is-referenced-by-count":3,"title":["Real functions on the family of all well-ordered subsets of a partially ordered set"],"prefix":"10.1017","volume":"48","author":[{"given":"Stevo","family":"Todor\u010devi\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200043383_ref007","volume-title":"Skolska knjiga","author":"Kurepa","year":"1951"},{"key":"S0022481200043383_ref002","article-title":"On a partition theorem of Baumgartner and Hajnal","volume":"10","author":"Galvin","journal-title":"Colloquia Mathematica Societatis Janos Bolyai"},{"key":"S0022481200043383_ref003","first-page":"1","article-title":"Ensembles ordonn\u00e9s et ramifi\u00e9s","volume":"4","author":"Kurepa","year":"1935","journal-title":"Publications Math\u00e9matiques de l'Universit\u00e9 de Belgrade"},{"key":"S0022481200043383_ref011","first-page":"A","article-title":"Real functions on the family of all well ordered subsets of a set","volume":"1","author":"Todor\u010devi\u0107","year":"1980","journal-title":"Abstracts of Papers Presented to the American Mathematical Society"},{"key":"S0022481200043383_ref008","first-page":"9","article-title":"Fonctions croissantes dans la famille des ensembles bien ordonn\u00e9s lin\u00e9aires","volume":"2","author":"Kurepa","year":"1954","journal-title":"Bulletin Scientifique Yougoslave"},{"key":"S0022481200043383_ref005","first-page":"1033","article-title":"Transformations monotones des ensembles partiellement ordonn\u00e9s","volume":"205","author":"Kurepa","year":"1937","journal-title":"Comptes Rendus Hebdomadaires des S\u00e9ances de l'Acad\u00e9mic des Sciences (Paris)"},{"key":"S0022481200043383_ref006","first-page":"827","article-title":"Transformations monotones des ensembles partiellement ordonn\u00e9s","volume":"434","author":"Kurepa","year":"1940","journal-title":"Revista da Ciencias (Lima)"},{"key":"S0022481200043383_ref001","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.67.4.1748"},{"key":"S0022481200043383_ref009","first-page":"35","article-title":"Sur les fonctions r\u00e9elles dans la famille des ensembles bien ordonn\u00e9s de nombres rationels","volume":"4","author":"Kurepa","year":"1954","journal-title":"Bulletin International de l'Acad\u00e9mic Yougoslave des Sciences et des Beaux-Arts"},{"key":"S0022481200043383_ref010","first-page":"175","article-title":"Monotone mappings between some kinds of ordered sets","volume":"19","author":"Kurepa","year":"1964","journal-title":"Glasnik Matematicko-Fizi\u010dki i Astronomski"},{"key":"S0022481200043383_ref012","first-page":"A","article-title":"A partition relation for partially ordered sets","volume":"2","author":"Todor\u010devi\u0107","year":"1981","journal-title":"Abstracts of Papers Presented to the American Mathematical Society"},{"key":"S0022481200043383_ref004","first-page":"129","article-title":"Ensembles lin\u00e9aires et une classe de tableaux ramifi\u00e9s (Tableaux ramifi\u00e9s de M. Aronszajn)","volume":"6","author":"Kurepa","year":"1937","journal-title":"Publications Math\u00e9matiques de l'Universit\u00e9 de Belgrade"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200043383","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T19:36:38Z","timestamp":1558726598000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200043383\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,3]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1983,3]]}},"alternative-id":["S0022481200043383"],"URL":"https:\/\/doi.org\/10.2307\/2273324","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,3]]}}}