发表机构
Romanian Academy; A.I.Cuza University（罗马尼亚科学院; 阿伊库扎大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有限支撑结构提出有限依赖轮廓与不变性层级，揭示交律、并律等性质与最小支撑、轨道有限性及虚设消去等概念的联系。
AI 中文摘要
有限支撑表明依赖是有限的，但并未说明每个有限上下文如何控制可观察的内容。对于数据对称群 G \leq Sym(A) 和有限支撑的 G-集合 X，我们研究了由有限上下文 S \subseteq A 索引的有限依赖轮廓 X^S = {x: Fix_G(S) 固定 x}。对于 A^n 上的关系，这产生了布尔代数 B^{(n)}_{G,S} = Inv_n(Fix_G(S))，形成不变性层级。该层级区分了若干名义原则。其交律是 Bojanczyk-Klin-Lasota 最小支撑准则的关系代数形式，而层内单射性则是可替换性（fungibility）。独立并律控制跨上下文并集的复合；它对于纯关系语言（元数至多为二）中超齐次结构的局部寡态自同构群成立，但对于三元数据上的一元观察可能失效。新鲜性按种类逐一保持有效，其无限制的 some/any 形式在闭群中刻画了完全相等对称性。在最小支撑下，轨道有限性蕴含有限上下文层级和一致支撑界；在局部寡态性下，其逆命题也成立。逐点闭包不改变轮廓，标准原子位点表示表明，具有相同年龄的适当超齐次结构具有等价的作用拓扑斯，尽管它们的实质 FSS 宇宙可能保留不同的外部基数数据。上下文可定向性给出了一个互补的点状不变量，其阈值为从无序对中选择的最小支撑大小。对于具有退化代数闭包的可数 omega-范畴结构，交律等价于虚设的弱消去。
英文摘要
Finite support indicates that the dependence is limited, but not how each finite context controls what can be observed. For a data symmetry G \leq Sym(A) and a finitely supported G-set X, we study the finite-dependence profile X^S = {x : Fix_G(S) fixes x}, indexed by finite contexts S \subseteq A. For relations on A^n this yields Boolean algebras B^{(n)}_{G,S} = Inv_n(Fix_G(S)), forming the invariance hierarchy. The hierarchy separates several nominal principles. Its meet law is the relation-algebra form of the Bojanczyk-Klin-Lasota least-support criterion, while level injectivity is fungibility. The independent join law governs composition across unions of contexts; it holds for locally oligomorphic automorphism groups of ultrahomogeneous structures in purely relational languages of arity at most two, but can fail for unary observations over ternary data. Freshness remains valid sort by sort, and its unrestricted some/any form characterizes full equality symmetry among closed groups. With least supports, orbit-finiteness implies finite context levels and a uniform support bound; under local oligomorphicity the converse also holds. Pointwise closure does not change the profile, and standard atomic-site presentations show that suitable ultrahomogeneous structures with the same age have equivalent action topoi, even though their material FSS universes may retain different external cardinal data. Contextual orientability gives a complementary pointed invariant whose threshold is the least support size of choice from unordered pairs. For countable omega-categorical structures with degenerate algebraic closure, the meet law is equivalent to weak elimination of imaginaries.
Comments44 pages, 2 tables, no figures. Unified synthesis of finite-dependence profiles, invariance hierarchies, freshness, context composition, orbit-finiteness, and contextual orientability
