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March 2004 Approximate Euler characteristic, dimension, and weak pigeonhole principles
Jan Krajíček
J. Symbolic Logic 69(1): 201-214 (March 2004). DOI: 10.2178/jsl/1080938837

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 WPHP2nn: 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 WPHP2nn.

Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle WPHPn2n: for no definable set A with more than one element can A2 definably embed into A.

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Jan Krajíček. "Approximate Euler characteristic, dimension, and weak pigeonhole principles." J. Symbolic Logic 69 (1) 201 - 214, March 2004. https://doi.org/10.2178/jsl/1080938837

Information

Published: March 2004
First available in Project Euclid: 2 April 2004

zbMATH: 1068.03024
MathSciNet: MR2039357
Digital Object Identifier: 10.2178/jsl/1080938837

Rights: Copyright © 2004 Association for Symbolic Logic

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Vol.69 • No. 1 • March 2004
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