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