A Special $Π^0_1$ Class with the Join Property but without Pseudojump Inversion

Abstract

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.