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Strongly and co-strongly minimal abelian structures

Published online by Cambridge University Press:  12 March 2014

Ehud Hrushovski  and
James Loveys
Ehud Hrushovski
Affiliation:
Department of Mathematicc, Hebrew University at Jerusalem, 91904 Jerusalem, Israel. E-mail: ehud@math.huji.ac.il
James Loveys
Affiliation:
The Department of Mathematics and Statistics, Mcgill University, Burnside Hall, Room 916, 805 Sherbrooke W. Montreal, Qc, H3A 2K6, Canada. E-mail: loveys@math.mcgill.ca
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Abstract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 2 , June 2010 , pp. 442 - 458
Copyright
Copyright © Association for Symbolic Logic 2010

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References

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