Let \(\widetilde{{\mathcal {M}}}=\langle {{{\mathcal {M}}}}, G\rangle \) be an expansion of a real closed field \({{{\mathcal {M}}}}\) by a dense subgroup G of \(\langle M^{>0}, \cdot \rangle \) with the Mann property. We prove that the induced structure on G by \({{{\mathcal {M}}}}\) eliminates imaginaries. As a consequence, every small set X definable in \({{{\mathcal {M}}}}\) can be definably embedded into some \(G^l\), uniformly in parameters. These results are proved in a more general setting, where \(\widetilde{{\mathcal {M}}}=\langle {{{\mathcal {M}}}}, P\rangle \) is an expansion of an o-minimal structure \({{\mathcal {M}}}\) by a dense set \(P\subseteq M\), satisfying three tameness conditions.

令\（\ widetilde {{\ mathcal {M}}} = \ langle {{{\ mathcal {M}}}}，G \ rangle \）是实封闭字段\（{{{\ mathcal {中号}}}} \）由致密的子组G ^的\（\ langle M 1 {> 0}，\ CDOT \ rangle \）用Mann属性。我们证明了该感应结构ģ通过\（{{{\ mathcal {M}}}} \）消除imaginaries。结果，可以将\（{{{\ mathcal {M}}}} \）中定义的每个小集合X定义为均匀地嵌入参数中的某些\（G ^ l \）中。这些结果在更一般的设置中得到证明，其中\（\ widetilde {{\ mathcal {M}}} = \ langle {{{\ mathcal {M}}}}，P \ rangle \）是o最小结构\（{{\ mathcal {M}}} \）通过密集集\（P \ subseteq M \）的扩展，满足三个驯服条件。