Combinatorics with definable sets: Euler characteristics and grothendieck rings

Bulletin of Symbolic Logic 6 (3):311-330 (2000)
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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

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Citations of this work

Weakly o-Minimal Nonvaluational Structures.Roman Wencel - 2008 - Annals of Pure and Applied Logic 154 (3):139-162.
O-Minimalism.Hans Schoutens - 2014 - Journal of Symbolic Logic 79 (2):355-409.
A Remark on Divisibility of Definable Groups.Mário J. Edmundo - 2005 - Mathematical Logic Quarterly 51 (6):639-641.
Grothendieck Rings of ℤ-Valued Fields.Raf Cluckers & Deirdre Haskell - 2001 - Bulletin of Symbolic Logic 7 (2):262-269.
Grothendieck Rings of Theories of Modules.Amit Kuber - 2015 - Annals of Pure and Applied Logic 166 (3):369-407.

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References found in this work

Existence and Feasibility in Arithmetic.Rohit Parikh - 1971 - Journal of Symbolic Logic 36 (3):494-508.
A Mathematical Incompleteness in Peano Arithmetic.Jeff Paris & Leo Harrington - 1977 - In Jon Barwise & H. Jerome Keisler (eds.), Handbook of Mathematical Logic. North-Holland Pub. Co.. pp. 90--1133.

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