Combinatorics with definable sets: Euler characteristics and grothendieck rings

Bulletin of Symbolic Logic 6 (3):311-330 (2000)
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
Keywords First Order Structure   Euler Characteristic   Grothendieck Ring
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DOI 10.2307/421058
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References found in this work BETA
Rohit Parikh (1971). Existence and Feasibility in Arithmetic. Journal of Symbolic Logic 36 (3):494-508.

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Citations of this work BETA
Roman Wencel (2008). Weakly o-Minimal Nonvaluational Structures. Annals of Pure and Applied Logic 154 (3):139-162.
Amit Kuber (2015). Grothendieck Rings of Theories of Modules. Annals of Pure and Applied Logic 166 (3):369-407.
Hans Schoutens (2014). O-Minimalism. Journal of Symbolic Logic 79 (2):355-409.

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