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Weakly minimal groups of unbounded exponent

Published online by Cambridge University Press:  12 March 2014

James Loveys
James Loveys*
Affiliation:
Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 6K6, Canada
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Extract

Weakly minimal sets were first invented by Shelah [S] to solve Łoś' conjecture for uncountable theories. The first major result about them, which is that any nonalgebraic strong type either is of Morley rank 1 or has locally modular geometry, was proved by Buechler [B1]. Recently, Hrushovski [Hr] showed that a locally modular weakly minimal set interprets an abelian group, a result which revolutionized the study of these sets. Also Hrushovski showed that the geometry on the connected component of a weakly minimal locally modular group is that of a vector space over a division ring. This is also implicit in the result of Pillay and Hrushovski (Theorem 1.3 from [HP]) quoted below. In a sense, this completes their study, as any vector space over any division ring is in fact strongly minimal if there is no other structure. But the question remains as to what other structure such a group could have. This is the issue we address here.

Some work has been done on this question; in fact, Pillay ([Pi], generalized in [HP]) has demonstrated, in the more general context of a weakly normal group A, that any definable subset of An is in fact a Boolean combination of almost 0-definable cosets of almost 0-definable subgroups. In the weakly minimal case this can be sharpened a fair bit. The case of a weakly minimal group of bounded exponent is dealt with in [HL]. Here we consider the case where the group has unbounded exponent.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 3 , September 1990 , pp. 928 - 937
Copyright
Copyright © Association for Symbolic Logic 1990

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References

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