{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,4]],"date-time":"2023-11-04T14:43:50Z","timestamp":1699109030921},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":3571,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,6]]},"abstract":"<jats:title>Abstract.<\/jats:title><jats:p>A relation on a linearly ordered structure is called semi-bounded if it is definable in an expansion of the structure by bounded relations. We study ultimate behavior of semi-bounded relations in an ordered module <jats:italic>M<\/jats:italic> over an ordered commutative ring <jats:italic>R<\/jats:italic> such that <jats:italic>M\/rM<\/jats:italic> is finite for all nonzero <jats:italic>r<\/jats:italic> \u03f5 <jats:italic>R<\/jats:italic>. We consider <jats:italic>M<\/jats:italic> as a structure in the language of ordered <jats:italic>R<\/jats:italic>-modules augmented by relation symbols for the submodules <jats:italic>rM<\/jats:italic>, and prove several quantifier elimination results for semi-bounded relations and functions in <jats:italic>M<\/jats:italic>. We show that these quantifier elimination results essentially characterize the ordered modules <jats:italic>M<\/jats:italic> with finite indices of the submodules <jats:italic>rM<\/jats:italic>. It is proven that (1) any semi-bounded <jats:italic>k<\/jats:italic>-ary relation on <jats:italic>M<\/jats:italic> is equal, outside a finite union of <jats:italic>k<\/jats:italic>-strips, to a <jats:italic>k<\/jats:italic>-ary relation quantifier-free definable in <jats:italic>M<\/jats:italic>, (2) any semibounded function from <jats:italic>M<\/jats:italic><jats:sup><jats:italic>k<\/jats:italic><\/jats:sup> to <jats:italic>M<\/jats:italic> is equal, outside a finite union of <jats:italic>k<\/jats:italic>-strips, to a piecewise linear function, and (3) any semi-bounded in <jats:italic>M<\/jats:italic> endomorphism of the additive group of <jats:italic>M<\/jats:italic> is of the form <jats:italic>x<\/jats:italic> \u21a6 \u03c3<jats:italic>x<\/jats:italic>, for some \u03c3 from the field of fractions of <jats:italic>R<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1082418540","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:37:03Z","timestamp":1109799423000},"page":"499-517","source":"Crossref","is-referenced-by-count":1,"title":["Semi-bounded relations in ordered modules"],"prefix":"10.1017","volume":"69","author":[{"given":"Oleg","family":"Belegradek","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007878_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761295"},{"key":"S0022481200007878_ref003","unstructured":"Edmundo M\u00e1rio Jorge , O-minimal expansions of groups, Ph.D. thesis, University of Oxford, 1999."},{"key":"S0022481200007878_ref004","first-page":"138","volume-title":"The model theory of groups","author":"Herzog","year":"1989"},{"key":"S0022481200007878_ref006","first-page":"779","volume":"57","author":"Peterzil","year":"1992","journal-title":"A structure theorem for semibounded sets in the reals"},{"key":"S0022481200007878_ref001","volume-title":"Semi-bounded relations in ordered abelian groups","author":"Belegradek","year":"2002"},{"key":"S0022481200007878_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(02)00084-2"},{"key":"S0022481200007878_ref007","doi-asserted-by":"publisher","DOI":"10.1216\/RMJ-1989-19-3-871"},{"key":"S0022481200007878_ref008","unstructured":"Poston Robert J. , O-minimal expansions of additive reals: definability of multiplication and some structural analysis, Ph.D. thesis, University of Oxford, 1999."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200007878","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T21:02:35Z","timestamp":1557176555000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200007878\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,6]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2004,6]]}},"alternative-id":["S0022481200007878"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1082418540","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,6]]}}}