带凸关系的实数集上的无穷本原正可定义性
Infinitary primitive positive definability over the real numbers with convex relations
AI总结：
本研究针对带凸关系的实数结构，证明其上无穷本原正可定义关系的集合依凸集性质分为六类，给出几何与线性映射描述，并得出仿射与凸组合克隆间无局部闭克隆的结论。
摘要：
在有限结构上，多态不变关系恰好是本原正可定义的关系。在无限结构上，这两类关系通常并不相同。无穷本原正可定义关系是一个自然的中间概念，它通过无限合取扩展了本原正可定义性。\n 我们对每个凸集$S\subset \mathbb{R}^n$，考虑带有加法、标量乘法、常元以及额外关系$S$的实数结构$\mathbb{R}$。我们证明，取决于$S$的不同，该结构中所有具有无穷本原正定义的关系的集合等于六种可能集合中的一种。这种依赖关系将凸集自然划分为六个非空类。我们还给出了这些类的初等几何描述，以及基于线性映射的描述。\n 该分类还表明，在仿射组合克隆与凸组合克隆之间不存在局部闭克隆。
英文摘要：
On a finite structure, the polymorphism invariant relations are exactly the primitively positively definable relations. On infinite structures, these two sets of relations are different in general. Infinitary primitively positively definable relations are a natural intermediate concept which extends primitive positive definability by infinite conjunctions. We consider for every convex set $S\subset \mathbb{R}^n$ the structure of the real numbers $\mathbb{R}$ with addition, scalar multiplication, constants, and additionally the relation $S$. We prove that depending on $S$, the set of all relations with an infinitary primitive positive definition in this structure equals one out of six possible sets. This dependency gives a natural partition of the convex sets into six nonempty classes. We also give an elementary geometric description of the classes and a description in terms of linear maps. The classification also implies that there is no locally closed clone between the clone of affine combinations and the clone of convex combinations.
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