实用与结构性的无穷逻辑扩张
Practical and Structural Infinitary Expansions
AI总结：
本文引入了无穷逻辑扩张，证明其保持并反映嵌入 Ramsey 性质和建模性质，去除了 Scow 定理的技术假设，并强化了 Kechris-Pestov-Todorcevic 对应。
摘要：
给定一个结构 $M$，我们引入了无穷逻辑扩张（infinitary logic expansions），它推广了 Morleyisation。我们证明了这些扩张是温顺的，即它们保持并反映了嵌入 Ramsey 性质（ERP）和建模性质（MP）。随后我们将注意力转向 Scow 定理，该定理将广义不可辨元与 Ramsey 类联系起来，并表明通过无穷逻辑，可以得到一个更强的结果，该结果不需要任何技术性假设。我们还证明了每个具有 ERP 的结构（不一定是可数的）都允许一个线性序，该序是无量词类型的并集，从而有效地证明了任何 Ramsey 结构“本质上是”有序的。我们还为不一定是有限的结构类引入了 ERP 的一个版本（finitary-ERP），并证明了针对该概念的 Kechris-Pestov-Todorcevic 对应的强化形式。
英文摘要：
Given a structure $M$ we introduce infinitary logic expansions, which generalise the Morleyisation. We show that these expansions are tame, in the sense that they preserve and reflect both the Embedding Ramsey Property (ERP) and the Modelling Property (MP). We then turn our attention to Scow's theorem connecting generalised indiscernibles with Ramsey classes and show that by passing through infinitary logic, one can obtain a stronger result, which does not require any technical assumptions. We also show that every structure with ERP, not necessarily countable, admits a linear order which is a union of quantifier-free types, effectively proving that any Ramsey structure is ``essentially'' ordered. We also introduce a version of ERP for classes of structures which are not necessarily finite (the finitary-ERP) and prove a strengthening of the Kechris-Pestov-Todorcevic correspondence for this notion.
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