Journal of Symbolic Logic

Groups definable in ordered vector spaces over ordered division rings

Pantelis E. Eleftheriou and Sergei Starchenko

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Abstract

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

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

Dates
First available in Project Euclid: 18 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1203350776

Digital Object Identifier
doi:10.2178/jsl/1203350776

Mathematical Reviews number (MathSciNet)
MR2371195

Zentralblatt MATH identifier
1130.03028

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

Keywords
O-minimal structures definably compact groups quotient by lattice

Citation

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. https://projecteuclid.org/euclid.jsl/1203350776


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