{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T19:59:13Z","timestamp":1699905553136},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7862,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1992,9]]},"abstract":"<jats:p>In [MPP] it was shown that in every reduct <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline1\" \/> of <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline2\" \/> = \u2039 \u211d, +, \u00b7, &lt;\u203a that properly expands <jats:italic>\u2133<\/jats:italic> = \u2039\u211d, +, &lt;, <jats:italic>\u03bb<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub>\u203a<jats:sub><jats:italic>a<\/jats:italic>\u2208\u211d<\/jats:sub>, all the bounded semi-algebraic (that is, <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline2\" \/>-definable) sets are definable. Said differently, every such <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline1\" \/> is an expansion of <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline3\" \/> = \u2039\u211d, +, &lt;, <jats:italic>\u03bb<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub>, <jats:italic>B<jats:sub>i<\/jats:sub><\/jats:italic>\u203a<jats:sub><jats:italic>a<\/jats:italic>\u2208\u211d, <jats:italic>i<\/jats:italic>\u2208<jats:italic>I<\/jats:italic><\/jats:sub> where {<jats:italic>B<jats:sub>i<\/jats:sub><\/jats:italic>}<jats:sub><jats:italic>i<\/jats:italic>\u2208<jats:italic>I<\/jats:italic><\/jats:sub> is the collection of all bounded semialgebraic sets and the <jats:italic>\u03bb<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub>'s are scalar multiplication by <jats:italic>a<\/jats:italic>. In [PSS] (see Theorem 1.2 below) it was shown that the structure <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline3\" \/> is a proper reduct of <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline2\" \/>; that is, one cannot define in it all the semialgebraic sets. In [Pe] we show that <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline3\" \/> is the only reduct properly between <jats:italic>\u2133<\/jats:italic> and <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline2\" \/>. As a first step towards this result, we investigate in this paper the definable sets in reducts such as <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline3\" \/>. (We point out that \u2018definable\u2019 will always mean \u2018definable with parameters\u2019.)<\/jats:p><jats:p>Definition 1.1. Let <jats:italic>X<\/jats:italic> \u2286 \u211d<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>. <jats:italic>X<\/jats:italic> is called <jats:italic>semi-bounded<\/jats:italic> if it is definable in the structure \u2039\u211d, +, &lt;, <jats:italic>\u03bb<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub>, <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>B<jats:sub>k<\/jats:sub><\/jats:italic>\u203a<jats:sub><jats:italic>a<\/jats:italic>\u2208\u211d<\/jats:sub>, where the <jats:italic>B<jats:sub>i<\/jats:sub><\/jats:italic>'s are bounded subsets of \u211d<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>.<\/jats:p><jats:p>The main result of this paper (see Theorem 3.1) shows roughly that, in Ominimal expansions of <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022349_inline2\" \/> that satisfy the partition condition (see Definition 2.3), every semibounded set can be partitioned into finitely many sets, each of which is of a form similar to a cylinder. Namely, these sets are obtained through the \u201cstretching\u201d of a bounded cell by finitely many linear vectors. As a corollary (see Theorem 1.4), we get different characterizations of semibounded sets, either in terms of their structure or in terms of their definability power.<\/jats:p><jats:p>The following result, by A. Pillay, P. Scowcroft and C. Steinhorn, was the main motivation for this paper. The theorem is formulated here in a slightly stronger form than originally, but the proof itself is essentially the original one. A short version of the proof is included in \u00a74.<\/jats:p>","DOI":"10.2307\/2275430","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:45:35Z","timestamp":1146955535000},"page":"779-794","source":"Crossref","is-referenced-by-count":15,"title":["A structure theorem for semibounded sets in the reals"],"prefix":"10.1017","volume":"57","author":[{"given":"Ya'acov","family":"Peterzil","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200022349_ref003","first-page":"109","volume":"57","author":"Marker","year":"1992","journal-title":"Additive reducts of real closed fields"},{"key":"S0022481200022349_ref008","first-page":"796","volume":"53","author":"van den Dries","year":"1988","journal-title":"On the elementary theory of restricted analytic functions"},{"key":"S0022481200022349_ref007","doi-asserted-by":"publisher","DOI":"10.1216\/RMJ-1989-19-3-871"},{"key":"S0022481200022349_ref001","volume-title":"G\u00e9om\u00e9trie alg\u00e9brique r\u00e9elle","author":"Bochnak","year":"1988"},{"key":"S0022481200022349_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(88)90125-9"},{"key":"S0022481200022349_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0833698-1"},{"key":"S0022481200022349_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0833697-X"},{"key":"S0022481200022349_ref004","unstructured":"Peterzil Y. , Some definability questions in structures over the reals and in general O-minimal structures, Ph.D. thesis, University of California, Berkeley, California, 1991."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200022349","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T21:01:18Z","timestamp":1558040478000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/product\/identifier\/S0022481200022349\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,9]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1992,9]]}},"alternative-id":["S0022481200022349"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.2307\/2275430","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,9]]}}}