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Generalized Tukey reducibility between σ-directed sets
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DOI：10.1007/s00153-026-01019-1.png)
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Abstract
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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 ZF (without the axiom of choice) to compare cofinal types of directed sets. In 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), <=*), (M, subset of), (N, subset of), ([omega(omega)](omega), subset of) and (omega(1), <=) where <=* denotes the dominating relation on omega omega, and M and N denote the ideal of meager and null sets of 2(omega), respectively. We show that under ZF + 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(R) satisfying the axiom of determinacy.
Keywords:
Tukey reducibility
Tukey map
Convergent map
Directed sets
Axiom of Choice
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