Sciweavers

BSL
2000

Combinatorics with definable sets: Euler characteristics and Grothendieck rings

13 years 5 months ago
Combinatorics with definable sets: Euler characteristics and Grothendieck rings
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 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 counterexamples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.
Jan Krajícek, Thomas Scanlon
Added 17 Dec 2010
Updated 17 Dec 2010
Type Journal
Year 2000
Where BSL
Authors Jan Krajícek, Thomas Scanlon
Comments (0)