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June 2010 Strongly and co-strongly minimal abelian structures
Ehud Hrushovski, James Loveys
J. Symbolic Logic 75(2): 442-458 (June 2010). DOI: 10.2178/jsl/1268917489


We give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:

  • 1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);

  • 2. when the theory of the structure is strongly minimal.

In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d∈ D, the index of A∩ dA in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.


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Ehud Hrushovski. James Loveys. "Strongly and co-strongly minimal abelian structures." J. Symbolic Logic 75 (2) 442 - 458, June 2010.


Published: June 2010
First available in Project Euclid: 18 March 2010

zbMATH: 1197.03040
MathSciNet: MR2648150
Digital Object Identifier: 10.2178/jsl/1268917489

Rights: Copyright © 2010 Association for Symbolic Logic


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Vol.75 • No. 2 • June 2010
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