发表机构
University of Camerino（卡梅里诺大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了一个具有并性质但不具有伪跳跃逆性质的特殊 $\Pi^0_1$ 类，从而完成四种存在性图景，并证明相关类可限于广义低实数且并性质局部成立。
AI 中文摘要
Jananthan 和 Simpson 表明，在由并性质与伪跳跃逆性质是否成立所决定的四种可能情形中，特殊 $\Pi^0_1$ 类存在于其中三种，而具有并性质但不具有伪跳跃逆性质的特殊 $\Pi^0_1$ 类的存在性仍为开放问题。我们构造了这样一个类，从而肯定地回答了他们的提问，并完成了四种情形的存在性图景。此外，该类可以被选取为完全由广义低实数组成，并且并性质在每个非空相对基本开子类上局部成立。更一般地，包含在 $\mathrm{GL}_1$ 中的每个特殊 $\Pi^0_1$ 类都有一个仍包含在 $\mathrm{GL}_1$ 中的特殊 $\Pi^0_1$ 扩张，且并性质在该扩张上局部成立。我们还证明了包含在 $\mathrm{GL}_1$ 中的任何类都不具有伪跳跃逆性质。一个独立的附录证明了，对于任何具有并性质的类，$0'$ 之上的每个跳跃逆纤维都是可数无穷的。
英文摘要
Jananthan and Simpson showed that special $Π^0_1$ classes exist in three of the four possible cases determined by whether the Join Property and the Pseudojump Inversion Property hold, leaving open the existence of a special $Π^0_1$ class with the Join Property but not the Pseudojump Inversion Property. We construct such a class, thereby answering their question affirmatively and completing the four-way existence pattern. Moreover, the class can be chosen to consist entirely of generalized low reals, with the Join Property holding locally on every nonempty relatively basic open subclass. More generally, every special $Π^0_1$ class contained in $\mathrm{GL}_1$ has a special $Π^0_1$ extension, still contained in $\mathrm{GL}_1$, on which the Join Property holds locally. We also show that no class contained in $\mathrm{GL}_1$ has the Pseudojump Inversion Property. An independent appendix proves that, for any class with the Join Property, every jump-inversion fibre above $0'$ is countably infinite.
Comments19 pages
