AI 中文总结
本文分离Madarász定理的仿射几何成分，提出半线性忠实性假设，扩展有限关系坐标几何的可定义性刻画，其框架可应用于纯域外的多种几何场景。
AI 中文摘要
Madarász近期的定理通过有限域可定义坐标几何在其仿射自同构下的不变性，刻画了无参数概念。本文分离该证明中的仿射几何成分，将论证扩展至包含额外有限关系坐标几何。新假设为“半线性忠实性”：在每个超幂中，诱导几何的每个自同构可分解为几何的仿射自同构，再跟随扩展基结构自同构的分量作用。在此假设下，每个无参数环境可定义关系恰好是几何的概念当且仅当它被仿射自同构群保持，还得到了对应概念集的对偶包含定理。该框架可恢复原始有限域可定义定理，且适用于纯域可定义之外的场景，包括命名标量几何和指数域上的框架扩展几何。最后一节记录了任意（含非几何）单类结构中一致编码对称群的类似因子化原理。
英文摘要
A recent theorem of Madarasz characterises the parameter-free concepts of a finitely field-definable coordinate geometry by invariance under its affine automorphisms. This paper isolates the affine-geometric ingredient in that proof and extends the argument to include additional finite-relational coordinate geometries. The new hypothesis is "semilinear faithfulness": in every ultrapower, each automorphism of the induced geometry factors as an affine automorphism of the geometry followed by the componentwise action of an automorphism of the expanded base structure. Under this hypothesis, every parameter-free ambient-definable relation is a concept of the geometry exactly when it is preserved by the affine automorphism group. A corresponding dual inclusion theorem is obtained for concept sets. The framework recovers the original finitely field-definable theorem and applies beyond pure-field definability, including a named-scalar geometry and a frame-expanded geometry over an exponential field. A final section records the analogous factorisation principle for uniformly coded symmetry groups in arbitrary, including non-geometric, one-sorted structures.
CommentsA preliminary version of this paper was given as a talk, "Why affine geometry?", presented at LRB'2026, Budapest, July 2026
