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Approximate Euler characteristic, dimension, and weak pigeonhole principles

Published online by Cambridge University Press:  12 March 2014

Jan Krajíček
Jan Krajíček*
Affiliation:
Mathematical Institute, Academy of Sciences, Žitná 25, Prague 1, CZ-115 67, The Czech Republic, E-mail: krajicek@math.cas.cz
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Abstract

We define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle : two disjoint copies of a non-empty definable set A cannot be definably embedded into A, and principle CC of comparing cardinalities: for any two definable sets A, B either A definably embeds in B or vice versa. Also, a structure admitting a non-trivial approximate Euler characteristic must satisfy .

Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle : for no definable set A with more than one element can A2 definably embed into A.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 1 , March 2004 , pp. 201 - 214
Copyright
Copyright © Association for Symbolic Logic 2004

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References

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