Combinatorics with definable sets: Euler characteristics and Grothendieck rings
Combinatorics with definable sets: Euler characteristics and Grothendieck rings
复制标题
具有可定义集合的组合学：欧拉特征和格洛腾迪克环
DOI：
--
复制
发表时间：
2000
影响因子：
0.6
通讯作者：
T. Scanlon
中科院分区：
文献类型：
--
作者：
J. Krajícek;T. Scanlon
我们回顾了一阶结构的弱和强欧拉特征的概念，并明确了结构的Grothendieck环的概念。定义了偏序Euler特征线和Grothendieck环，给出了具有非平凡偏序Grothendieck环的结构的一个刻划。我们给出了一个推广的计数功能的局部有限结构，并使用的建设表明，Grothendieck环的复数包含作为一个子环的整数多项式环在连续多变量。证明了结构上存在泛强欧拉特征线.本文研究了Grothendieck环对结构理论的依赖性，并给出了几个反例。最后，我们将有界算术中的一些公开问题和独立性结果与特殊Grothendieck环的性质联系起来。
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.