We introduce a generalization of the Tukey reducibility, which we call the pre-Tukey reducibility. While the basics of the original Tukey reducibility heavily rely on the axiom of choice, the pre-Tukey reducibility works well in \(\textsf{ZF}\) (without the axiom of choice) to compare cofinal types of directed sets. In \(\textsf{ZF}\) , we show that two directed sets are pre-Tukey equivalent if and only if there exists a directed set into which both directed sets can be cofinally embedded. In this paper, we investigate the pre-Tukey reducibility between \(\sigma \) -directed sets \((\omega ^\omega , \le ^*), (\mathcal {M}, \subseteq ), (\mathcal {N}, \subseteq ), ([\omega ^\omega ]^\omega , \subseteq ) \) and \((\omega _1, \le )\) where \(\le ^*\) denotes the dominating relation on \(\omega ^\omega \) , and \(\mathcal {M} \) and \(\mathcal {N}\) denote the ideal of meager and null sets of \(2^\omega \) , respectively. We show that under \(\textsf{ZF}+ \textsf{DC}\) and certain assumptions on sets of reals, these directed sets have pairwise distinct cofinal types. The assumptions we consider hold in the Solovay model and in \(L(\mathbb {R})\) satisfying the axiom of determinacy.
