Journal of Symbolic Logic

Groups definable in ordered vector spaces over ordered division rings

Pantelis E. Eleftheriou and Sergei Starchenko

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Let $\mathcal{M} = \langle M, +, <, 0, \{\lambda\}_{\lambda \in D}\rangle$ be an ordered vector space over an ordered division ring $D$, and $G = \langle G, \oplus, e_G \rangle$ an $n$-dimensional group definable in $\mathcal{M}. We show that if $G$ is definably compact and definably connected with respect to the $t$-topology, then it is definably isomorphic to a ‘definable quotient group’ $U/L$, for some convex $\bigvee$-definable subgroup $U$ of $\langle M^n, + \rangle$ and a lattice $L$ of rank $n$. As two consequences, we derive Pillay’s conjecture for a saturated $\mathcal{M}$ as above and we show that the o-minimal fundamental group of $G$ is isomorphic to $L$.

Article information

J. Symbolic Logic, Volume 72, Issue 4 (2007), 1108-1140.

First available in Project Euclid: 18 February 2008

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C64: Model theory of ordered structures; o-minimality 22C05: Compact groups 46A40: Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42]

O-minimal structures definably compact groups quotient by lattice


Eleftheriou, Pantelis E.; Starchenko, Sergei. Groups definable in ordered vector spaces over ordered division rings. J. Symbolic Logic 72 (2007), no. 4, 1108--1140. doi:10.2178/jsl/1203350776.

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