# zbMATH — the first resource for mathematics

Lattices in locally definable subgroups of $$\langle R^{n},+\rangle$$. (English) Zbl 1345.03072
Summary: Let $$\mathcal{M}$$ be an o-minimal expansion of a real closed field $$R$$. We define the notion of a lattice in a locally definable group and then prove that every connected, definably generated subgroup of $$\langle R^{n},+\rangle$$ contains a definable generic set and therefore admits a lattice.

##### MSC:
 03C64 Model theory of ordered structures; o-minimality 03C68 Other classical first-order model theory 22B99 Locally compact abelian groups (LCA groups)
##### Keywords:
o-minimality; locally definable groups; lattices; generic sets
Full Text:
##### References:
 [1] Baro, E., and M. J. Edmundo, “Corrigendum to ‘Locally definable groups in o-minimal structures,”’ by M. J. Edmundo, Journal of Algebra , vol. 320 (2008), pp. 3079-80. [2] Baro, E., and M. Otero, “Locally definable homotopy,” Annals of Pure and Applied Logic , vol. 161 (2010), pp. 488-503. · Zbl 1225.03043 [3] Edmundo, M. J., “Locally definable groups in o-minimal structures,” Journal of Algebra , vol. 301(2006), pp. 194-223. · Zbl 1104.03032 [4] Eleftheriou, P. E., and Y. Peterzil, “Definable quotients of locally definable groups,” Selecta Mathematica (N.S.) , vol. 18 (2012), pp. 885-903. · Zbl 1273.03130 [5] Eleftheriou, P. E., and Y. Peterzil, “Definable groups as homomorphic images of semilinear and field-definable groups,” Selecta Mathematica (N.S.) , vol. 18 (2012), pp. 905-40. · Zbl 1273.03129 [6] Hrushovski, E., Y. Peterzil, and A. Pillay, “Groups, measures, and the NIP,” Journal of the American Mathematical Society , vol. 21 (2008), pp. 563-96. · Zbl 1134.03024 [7] Peterzil, Y., and S. Starchenko, “Definable homomorphisms of abelian groups in o-minimal structures,” Annals of Pure and Applied Logic , vol. 101 (2000), pp. 1-27. · Zbl 0949.03033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.