一个有界有限一度，其一一度恰好构成一个稠密线性序
A Bounded Finite-One Degree Whose One-One Degrees Form Exactly a Dense Linear Order
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中文总结 AI 辅助
本文构造一个有界有限一度，使其一一度恰好构成无端点稠密线性序，肯定回答开放问题，证明采用块均匀化与吸收到穷竭定理。
中文摘要 AI 辅助
我们构造一个集合 $U\leq_T\emptyset''$，其有界有限一度（按一一可归约性内部排序）恰好构成一个无端点的可数稠密线性序。更精确地说，\\[ \left( \{[B]_1:B\equiv_{\mathrm{bfo}}U\},\leq_1 \right) \cong (\mathbb Q,\leq). \\] 这一精确实现与先前两个结果形成对比。在早期工作中，构造了一个有界有限一度，它包含 $(\mathbb Q,\leq)$ 的一个副本，但同一度也包含一个无限反链的一一度，并且更一般地，嵌入所有可数偏序的副本；因此稠密链并未穷尽该度。在另一个方向上，$m$-刚性产生一个几乎必然和余稀疏的障碍：对于测度为 $1$ 且余稀疏的一类集合，相应的有界有限一度包含一个无限反链的一一度，因而不是线性有序的。本构造表明，尽管存在这种典型的负面行为，精确的稠密线性序仍可能发生。特别地，它肯定地回答了 Richter、Stephan 和 Zhang 的开放问题 3。证明有两个主要部分。一个块均匀化构造产生一个非柱形集合 $U$、一个弱二元塔 $(U_e)_{e\in\mathbb N}$、两种颜色的可计算储备以及一个基吸收性质。一个抽象的吸收到穷竭定理随后表明，$U$ 的有界有限一度中的每个成员都一一等价于某个有限自连接 $mU_e$，从而得到上述穷竭分类。
英文摘要
We construct a set $U\leq_T\emptyset''$ whose bounded finite-one degree, ordered internally by one-one reducibility, consists exactly of a countable dense linear order without endpoints. More precisely, \[ \left( \{[B]_1:B\equiv_{\mathrm{bfo}}U\},\leq_1 \right) \cong (\mathbb Q,\leq). \] This exact realization contrasts with two previous results. In earlier work, a bounded finite-one degree was constructed that contains a copy of $(\mathbb Q,\leq)$, but the same degree also contains an infinite antichain of one-one degrees and, more generally, embedded copies of all countable partial orders; thus the dense chain does not exhaust the degree. In a different direction, $m$-rigidity yields an almost-sure and comeager obstruction: for a measure-$1$ and comeager class of sets, the corresponding bounded finite-one degree contains an infinite antichain of one-one degrees and hence is not linearly ordered. The present construction shows that, despite this typical negative behaviour, exact dense linear order can occur. In particular, it gives an affirmative answer to Open Question~3 of Richter, Stephan, and Zhang. The proof has two main parts. A block homogenization construction produces a noncylindrical set $U$, a weak dyadic tower $(U_e)_{e\in\mathbb N}$, computable reservoirs of both colours, and a base absorption property. An abstract absorption-to-exhaustivity theorem then shows that every member of the bounded finite-one degree of $U$ is one-one equivalent to some finite autojoin $mU_e$, thereby yielding the exhaustive classification above.
发表机构
- School of Science and Technology, University of Camerino（卡梅里诺大学科学技术学院）
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