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Combinatorics with Definable Sets: Euler Characteristics and Grothendieck Rings

Published online by Cambridge University Press:  15 January 2014

Jan Krajíček  and
Thomas Scanlon
Jan Krajíček
Affiliation:
Mathematical Institute, Academy Of Sciences žItná 25, Prague 115 67, The Czech RepublicE-mail:[email protected]
Thomas Scanlon
Affiliation:
Department Of Mathematics, University Of CaliforniaBerkeley, Ca 94720-3840, USAE-mail:[email protected]
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Abstract

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 6 , Issue 3 , September 2000 , pp. 311 - 330
Copyright
Copyright © Association for Symbolic Logic 2000

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