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Logic"],"published-print":{"date-parts":[[2021,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\widetilde{{\\mathcal {M}}}=\\langle {{{\\mathcal {M}}}}, G\\rangle $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mover>\n                      <mml:mi>M<\/mml:mi>\n                      <mml:mo>~<\/mml:mo>\n                    <\/mml:mover>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>\u27e8<\/mml:mo>\n                      <mml:mi>M<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>G<\/mml:mi>\n                      <mml:mo>\u27e9<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be an expansion of a real closed field <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\\mathcal {M}}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> by a dense subgroup <jats:italic>G<\/jats:italic> of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\langle M^{&gt;0}, \\cdot \\rangle $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>\u27e8<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>M<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>&gt;<\/mml:mo>\n                        <mml:mn>0<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u00b7<\/mml:mo>\n                    <mml:mo>\u27e9<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with the Mann property. We prove that the induced structure on <jats:italic>G<\/jats:italic> by <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\\mathcal {M}}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> eliminates imaginaries. As a consequence, every small set <jats:italic>X<\/jats:italic> definable in <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\\mathcal {M}}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> can be definably embedded into some <jats:inline-formula><jats:alternatives><jats:tex-math>$$G^l$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>G<\/mml:mi>\n                    <mml:mi>l<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, uniformly in parameters. These results are proved in a more general setting, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\widetilde{{\\mathcal {M}}}=\\langle {{{\\mathcal {M}}}}, P\\rangle $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mover>\n                      <mml:mi>M<\/mml:mi>\n                      <mml:mo>~<\/mml:mo>\n                    <\/mml:mover>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>\u27e8<\/mml:mo>\n                      <mml:mi>M<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mo>\u27e9<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is an expansion of an o-minimal structure <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\mathcal {M}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>M<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> by a dense set <jats:inline-formula><jats:alternatives><jats:tex-math>$$P\\subseteq M$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>P<\/mml:mi>\n                    <mml:mo>\u2286<\/mml:mo>\n                    <mml:mi>M<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, satisfying three tameness conditions.<\/jats:p>","DOI":"10.1007\/s00153-020-00747-2","type":"journal-article","created":{"date-parts":[[2020,9,2]],"date-time":"2020-09-02T21:02:42Z","timestamp":1599080562000},"page":"317-327","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Small sets in Mann pairs"],"prefix":"10.1007","volume":"60","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0687-1096","authenticated-orcid":false,"given":"Pantelis E.","family":"Eleftheriou","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,9,2]]},"reference":[{"key":"747_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.apal.2007.06.002","volume":"150","author":"A Berenstein","year":"2007","unstructured":"Berenstein, A., Ealy, C., G\u00fcnaydin, A.: Thorn independence in the field of real numbers with a small multiplicative group. 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