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ℵ0-categorical structures with arbitrarily fast growth of algebraic closure

Published online by Cambridge University Press:  12 March 2014

David M. Evans  and
M. E. Pantano
David M. Evans
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, England, E-mail: [email protected]
M. E. Pantano
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, England
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Extract

Various results have been proved about growth rates of certain sequences of integers associated with infinite permutation groups. Most of these concern the number of orbits of the automorphism group of an ℵ0-categorical structure on the set of unordered n-subsets or on the set of n-tuples of elements of . (Recall that by the Ryll-Nardzewski Theorem, if is countable and ℵ0-categorical, the number of the orbits of its automorphism group Aut() on the set of n-tuples from is finite and equals the number of complete n-types consistent with the theory of .) The book [Ca90] is a convenient reference for these results. One of the oldest (in the realms of ‘folklore’) is that for any sequence (Kn)n∈ℕ of natural numbers there is a countable ℵ0-categorical structure such that the number of orbits of Aut() on the set of n-tuples from is greater than kn for all n.

These investigations suggested the study of the growth rate of another sequence. Let be an ℵ0-categorical structure and X be a finite subset of . Let acl(X) be the algebraic closure of X, that is, the union of the finite X-definable subsets of . Equivalently, this is the union of the finite orbits on of Aut()(X), the pointwise stabiliser of X in Aut(). Define

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 3 , September 2002 , pp. 897 - 909
Copyright
Copyright © Association for Symbolic Logic 2002

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