{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T16:43:49Z","timestamp":1767199429308,"version":"build-2238731810"},"reference-count":14,"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":9507,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1988,3]]},"abstract":"<jats:p>\n                    A\n                    <jats:italic>dilator D<\/jats:italic>\n                    is a functor from ON to itself commuting with direct limits and pull-backs. A dilator\n                    <jats:italic>D<\/jats:italic>\n                    is a\n                    <jats:italic>flower<\/jats:italic>\n                    iff\n                    <jats:italic>D<\/jats:italic>\n                    (\n                    <jats:italic>x<\/jats:italic>\n                    ) is continuous. A flower\n                    <jats:italic>F<\/jats:italic>\n                    is\n                    <jats:italic>regular<\/jats:italic>\n                    iff\n                    <jats:italic>F<\/jats:italic>\n                    (\n                    <jats:italic>x<\/jats:italic>\n                    ) is strictly increasing and\n                    <jats:italic>F<\/jats:italic>\n                    (\n                    <jats:italic>f<\/jats:italic>\n                    )(\n                    <jats:italic>F<\/jats:italic>\n                    (\n                    <jats:italic>z<\/jats:italic>\n                    )) =\n                    <jats:italic>F<\/jats:italic>\n                    (\n                    <jats:italic>f<\/jats:italic>\n                    (\n                    <jats:italic>z<\/jats:italic>\n                    )) (for\n                    <jats:italic>f<\/jats:italic>\n                    \u03f5 ON(x,y),\n                    <jats:italic>z<\/jats:italic>\n                    \u03f5\n                    <jats:italic>X<\/jats:italic>\n                    ).\n                  <\/jats:p>\n                  <jats:p>\n                    Equalization is the following axiom: if\n                    <jats:italic>F, G<\/jats:italic>\n                    \u03f5 Fl\n                    <jats:sub>r<\/jats:sub>\n                    (class of regular flowers), then there is an\n                    <jats:italic>H<\/jats:italic>\n                    \u03f5 Fl\n                    <jats:sub>r<\/jats:sub>\n                    such that\n                    <jats:italic>F<\/jats:italic>\n                    \u00b0\n                    <jats:italic>H<\/jats:italic>\n                    =\n                    <jats:italic>G<\/jats:italic>\n                    \u00b0\n                    <jats:italic>H<\/jats:italic>\n                    . From this we can deduce that if\n                    <jats:italic>\u2131<\/jats:italic>\n                    is a set \u2286 Fl\n                    <jats:sub>r<\/jats:sub>\n                    , then there is an\n                    <jats:italic>H<\/jats:italic>\n                    \u03f5 Fl\n                    <jats:sub>r<\/jats:sub>\n                    which is the smallest equalizer of\n                    <jats:italic>\u2131<\/jats:italic>\n                    (it can be said that\n                    <jats:italic>H equalizes \u2131<\/jats:italic>\n                    iff for every\n                    <jats:italic>F, G<\/jats:italic>\n                    \u03f5\n                    <jats:italic>\u2131<\/jats:italic>\n                    we have\n                    <jats:italic>F<\/jats:italic>\n                    \u00b0\n                    <jats:italic>H<\/jats:italic>\n                    =\n                    <jats:italic>G<\/jats:italic>\n                    \u00b0\n                    <jats:italic>H<\/jats:italic>\n                    ). Equalization is not provable in set theory because equalization for denumerable flowers is equivalent to\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200028966_inline2.png\"\/>\n                    -determinacy (see a forthcoming paper by Girard and Kechris).\n                  <\/jats:p>\n                  <jats:p>\n                    Therefore it is interesting to effectively find, by elementary means, equalizers even in the simplest cases. The aim of this paper is to prove Girard and Kechris's conjecture: \u201c\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200028966_inline1.png\"\/>\n                    is the (smallest) equalizer for Fl\n                    <jats:sub>r<\/jats:sub>\n                    &lt;\n                    <jats:italic>\u03c9<\/jats:italic>\n                    \u201d (where Fl\n                    <jats:sub>r<\/jats:sub>\n                    &lt;\n                    <jats:italic>\u03c9<\/jats:italic>\n                    denotes the set of finite regular flowers). We will verify that\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200028966_inline1.png\"\/>\n                    is an equalizer of Fl\n                    <jats:sub>r<\/jats:sub>\n                    &lt;\n                    <jats:italic>\u03c9<\/jats:italic>\n                    ; we will sketch the proof that it is the smallest one at the end of the paper. We will denote\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200028966_inline1.png\"\/>\n                    by\n                    <jats:italic>H<\/jats:italic>\n                    .\n                  <\/jats:p>","DOI":"10.2307\/2274431","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:24:27Z","timestamp":1146939867000},"page":"105-123","source":"Crossref","is-referenced-by-count":1,"title":["Equalization of finite flowers"],"prefix":"10.1017","volume":"53","author":[{"given":"Stefano","family":"Berardi","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200028966_bib013","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71893-7"},{"key":"S0022481200028966_bib008","first-page":"713","volume":"49","author":"Girard","year":"1984","journal-title":"Functors and ordinal notations. 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