Bulletin of Symbolic Logic

Combinatorics with Definable Sets: Euler Characteristics and Grothendieck Rings

Jan Krajíček and Thomas Scanlon

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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.

Article information

Source
Bull. Symbolic Logic, Volume 6, Number 3 (2000), 311-330.

Dates
First available in Project Euclid: 20 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.bsl/1182353707

Mathematical Reviews number (MathSciNet)
MR1803636

Zentralblatt MATH identifier
0968.03036

JSTOR
links.jstor.org

Keywords
First Order Structure Euler Characteristic Grothendieck Ring

Citation

Krajíček, Jan; Scanlon, Thomas. Combinatorics with Definable Sets: Euler Characteristics and Grothendieck Rings. Bull. Symbolic Logic 6 (2000), no. 3, 311--330. https://projecteuclid.org/euclid.bsl/1182353707


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