{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T10:03:26Z","timestamp":1648980206450},"reference-count":9,"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>Supercompact cardinals are usually defined in terms of the existence of certain normal ultrafilters. It is well known that there is a natural partial ordering on the collection of all normal ultrafilters associated with a super-compact cardinal, that of normal ultrafilter restriction. Using this notion, we define a tree structure <jats:italic>T<\/jats:italic> on the collection of normal ultrafilters associated with a fixed supercompact cardinal. Many results already appearing in the literature can be conveniently phrased in terms of structural properties of <jats:italic>T<\/jats:italic> (see, e.g. [4] or [6]). In this paper, we establish additional structural facts concerning <jats:italic>T<\/jats:italic>.<\/jats:p><jats:p>In \u00a71 we standardize our notation and review some of the basic facts and methods that will be used throughout. \u00a72 begins with a presentation of an important technique, due to Solovay, which will be an important tool for us. Also in \u00a72, we begin a detailed study of the structure of <jats:italic>T<\/jats:italic> in terms of branching and the existence of many successors to branches at limit levels. \u00a73 contains results proving the existence of many nodes of <jats:italic>T<\/jats:italic> which do not have successors above certain levels of <jats:italic>T<\/jats:italic>. This complements work of Magidor [6] who established the existence of many nodes which have successors at all higher levels of <jats:italic>T<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273384","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:58:32Z","timestamp":1146952712000},"page":"89-109","source":"Crossref","is-referenced-by-count":0,"title":["Supercompact cardinals and trees of normal ultrafilters"],"prefix":"10.1017","volume":"47","author":[{"given":"Julius B.","family":"Barbanel","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120004473X_ref001","unstructured":"Barbanel J. B. , Results on supercompact cardinals, Ph.D. Thesis, State University of New York at Buffalo, 1979."},{"key":"S002248120004473X_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/025\/0379200"},{"key":"S002248120004473X_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(77)90088-3"},{"key":"S002248120004473X_ref007","first-page":"225","volume":"41","author":"Menas","year":"1976","journal-title":"A combinatorial property of P\u03ba(\u03bb)"},{"key":"S002248120004473X_ref002","volume-title":"Boolean-valued models and independence proofs in set theory","author":"Bell","year":"1977"},{"key":"S002248120004473X_ref003","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S002248120004473X_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90031-1"},{"key":"S002248120004473X_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0061131"},{"key":"S002248120004473X_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771583"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120004473X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T21:42:59Z","timestamp":1558734179000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120004473X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,3]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1982,3]]}},"alternative-id":["S002248120004473X"],"URL":"https:\/\/doi.org\/10.2307\/2273384","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,3]]}}}