Generalized Tukey reducibility between $\sigma$-directed sets

TL;DR

This paper introduces pre-Tukey reducibility, a generalized form of Tukey reducibility suitable for $ ext{ZF}$, and explores its properties among $\sigma$-directed sets under certain set-theoretic assumptions.

Contribution

It defines pre-Tukey reducibility and analyzes its behavior among $\sigma$-directed sets in models where specific set-theoretic assumptions hold.

Findings

Pre-Tukey reducibility extends Tukey reducibility to $ ext{ZF}$.

Analysis of reducibility in Solovay model and $L( ext{R})$ with $ ext{AD}.

Results show how reducibility relations vary under different set-theoretic assumptions.

Abstract

We introduce the pre-Tukey reducibility, a generalization of the Tukey reducibility between directed sets that works well in $\mathsf{ZF}$. We investigate the pre-Tukey reducibility between several $\sigma$-directed sets under assumptions on sets of reals, which hold in the Solovay model and in $L(\mathbb{R})$ satisfying $\mathsf{AD}$.