Pre-Tukey Reducibility in Directed Sets

• Pre-Tukey reducibility is defined by mapping a directed set into its completion of non-empty upward-closed subsets, generalizing classical Tukey reducibility without relying on choice.
• It utilizes multivalued pre-Tukey maps that send elements to nonempty subsets, preserving cofinality and recovering ordinary Tukey maps under the axiom of choice.
• The framework is applied to σ-directed orders, establishing strict reducibility patterns between structures like the meager and null ideals and clarifying cofinal similarity.

Searching arXiv for the most relevant papers on pre-Tukey reducibility and adjacent Tukey frameworks. Pre-Tukey reducibility is a generalization of Tukey reducibility between directed sets defined by passing from a directed set $E$ to its completion $E^*$ of non-empty upward-closed subsets ordered by reverse inclusion. It was introduced as a choice-free extension of Tukey reducibility that works well in $\mathsf{ZF}$, and it is designed to retain the cofinal-type perspective of classical Tukey theory while avoiding the need to select single witnesses from nonempty families (Sakai et al., 9 Jul 2025).

1. Definition and basic formalism

The underlying setting is the order theory of directed sets. A directed set is a poset in which every finite set has an upper bound, and a $\sigma$-directed set is a poset in which every countable set has an upper bound. For directed sets $(D,\le_D)$ and $(E,\le_E)$, classical Tukey theory uses either Tukey maps $f:D\to E$, which send unbounded sets to unbounded sets, or convergent maps $f:D\to E$, which send cofinal sets to cofinal sets (Sakai et al., 9 Jul 2025).

The new ingredient is the completion

$(D,\le_D)^*,$

defined as the directed set of all non-empty $\le_D$-upward closed subsets of $E^*$0, ordered by reverse inclusion. This completion carries a canonical cofinal embedding of $E^*$1 into $E^*$2 by

$E^*$3

Pre-Tukey reducibility is then defined by

$E^*$4

The associated equivalence relation is denoted $E^*$5 (Sakai et al., 9 Jul 2025).

A direct reformulation uses set-valued witnesses. A pre-Tukey map from $E^*$6 to $E^*$7 is a map $E^*$8 such that $E^*$9 for all $\mathsf{ZF}$0, and

$\mathsf{ZF}$1

Dually, a pre-convergent map from $\mathsf{ZF}$2 to $\mathsf{ZF}$3 is a map $\mathsf{ZF}$4 such that $\mathsf{ZF}$5 for all $\mathsf{ZF}$6, and

$\mathsf{ZF}$7

These formulations are equivalent to $\mathsf{ZF}$8 (Sakai et al., 9 Jul 2025).

2. Relation to ordinary Tukey reducibility and the role of choice

Pre-Tukey reducibility extends ordinary Tukey reducibility. If $\mathsf{ZF}$9 is a Tukey map, then

$\sigma$0

is a pre-Tukey map. Likewise, an ordinary convergent map yields a pre-convergent map by passing to upward closures (Sakai et al., 9 Jul 2025).

The distinction appears when witnesses are multivalued. If $\sigma$1 is a pre-Tukey map and one can choose a point $\sigma$2 for every $\sigma$3, then the chosen map $\sigma$4 is an ordinary Tukey map. The same selection principle converts a pre-convergent map into an ordinary convergent map. Consequently, under $\sigma$5, pre-Tukey reducibility and classical Tukey reducibility coincide (Sakai et al., 9 Jul 2025).

The motivation for the new notion is precisely that such selections need not exist in $\sigma$6. Pre-Tukey reducibility replaces single witnesses by non-empty sets of possible witnesses, thereby preserving the basic comparison framework without assuming global choice. The paper introducing the notion explicitly asks whether the coincidence of the Tukey relation and the pre-Tukey relation implies $\sigma$7 (Sakai et al., 9 Jul 2025).

3. Structural properties and cofinal similarity

Pre-Tukey reducibility is a preorder, and $\sigma$8 is an equivalence relation. Its main structural theorem is a choice-free analogue of Tukey’s cofinal-similarity theorem: two directed sets are pre-Tukey equivalent if and only if they are both cofinally embeddable into a common directed set (Sakai et al., 9 Jul 2025).

This result is significant because it shows that the new relation does not merely weaken Tukey reducibility. It still captures cofinal type in the sense that $\sigma$9 is exactly the equivalence relation of common cofinal embeddability, now proved without appealing to $(D,\le_D)$0 (Sakai et al., 9 Jul 2025).

A canonical example separates pre-Tukey reducibility from ordinary Tukey reducibility in $(D,\le_D)$1. Let $(D,\le_D)$2 be the reals coding well-orderings of $(D,\le_D)$3, ordered by $(D,\le_D)$4 via order type and then lexicographic refinement. If there is no injection from $(D,\le_D)$5 into $(D,\le_D)$6, then there is no Tukey map from $(D,\le_D)$7 to $(D,\le_D)$8. Nevertheless,

$(D,\le_D)$9

is a pre-Tukey map, and hence

$(E,\le_E)$0

The example isolates the exact role of multivalued witnesses: pre-Tukey reducibility can compare structures whose classical Tukey comparison would require choosing one code for each countable ordinal (Sakai et al., 9 Jul 2025).

4. Comparison theorems for $(E,\le_E)$1-directed orders

The principal applications concern the $(E,\le_E)$2-directed orders

$(E,\le_E)$3

where $(E,\le_E)$4 is the meager ideal on $(E,\le_E)$5 and $(E,\le_E)$6 is the null ideal on $(E,\le_E)$7. The ambient hypothesis is

$(E,\le_E)$8

a condition that holds in the Solovay model obtained from an inaccessible cardinal and in $(E,\le_E)$9 satisfying $f:D\to E$0 (Sakai et al., 9 Jul 2025).

Under $f:D\to E$1, the paper establishes the strict pattern

$f:D\to E$2

together with

$f:D\to E$3

The reverse arrows fail in the places indicated by the comparison diagram: in particular,

$f:D\to E$4

and $f:D\to E$5 is incomparable with each of $f:D\to E$6, $f:D\to E$7, and $f:D\to E$8 (Sakai et al., 9 Jul 2025).

A characteristic proof is the reduction

$f:D\to E$9

In $f:D\to E$0, such comparisons are often proved by choosing a particular code or decomposition of each meager set. The pre-Tukey proof avoids this by taking, for each meager set, the non-empty set of all compatible decompositions and using that family as the image of a pre-Tukey map. This is the paradigmatic use of the new notion: it replaces a non-canonical choice by a multivalued witness while preserving the desired cofinal comparison (Sakai et al., 9 Jul 2025).

The same paper also proves a general lemma for $f:D\to E$1-directed orders on $f:D\to E$2: if $f:D\to E$3 is $f:D\to E$4-directed, then

$f:D\to E$5

via the map sending a countable set $f:D\to E$6 to the set of all common upper bounds of $f:D\to E$7 (Sakai et al., 9 Jul 2025).

5. Place within the broader Tukey literature

The term “pre-Tukey reducibility” is highly specific. Most adjacent literatures do not use it, even when they introduce Tukey-like generalizations or refinements. In categorical Ramsey theory, the analogue is the notion of pre-adjunction, and a category $f:D\to E$8 is declared Tukey reducible to a category $f:D\to E$9 when there is a pre-adjunction from $(D,\le_D)^*,$0 to $(D,\le_D)^*,$1. On essentially countable directed preorders viewed as thin categories, this reproduces classical Tukey reducibility exactly (Barbosa et al., 2023).

In the theory of analytic directed orders, a different weakening appears: the relation $(D,\le_D)^*,$2, defined by requiring preimages of $(D,\le_D)^*,$3-bounded sets to be $(D,\le_D)^*,$4-bounded. It is explicitly weaker than $(D,\le_D)^*,$5 and preserves corresponding cardinal-invariant inequalities, but it is not the same notion as pre-Tukey reducibility (Solecki, 2015).

A different refinement occurs in the Vojtáš framework of relational systems. There the model-relative relation

$(D,\le_D)^*,$6

adds model-membership constraints to ordinary Tukey-style morphisms and is used to transfer statements of the form $(D,\le_D)^*,$7 and $(D,\le_D)^*,$8 (Cardona, 2021).

Ultrafilter theory provides another nearby landscape. Several papers in that area explicitly note that they do not introduce a notion called pre-Tukey reducibility; instead they compare ordinary Tukey reducibility with stronger orders such as Rudin–Keisler and Rudin–Blass reducibility. In particular, $(D,\le_D)^*,$9 is stronger than $\le_D$0, and on P-points the Rudin–Keisler and Rudin–Blass orders coincide [(Raghavan et al., 2014); (Dobrinen, 2014)]. These are precursor orders in a different sense: they are finer than Tukey reducibility, whereas pre-Tukey reducibility is a choice-free generalization of it.

6. Significance, strengthening phenomena, and open directions

Pre-Tukey reducibility is significant because it preserves the cofinal-type viewpoint of Tukey theory while removing the dependence on single-valued witnesses. It is therefore a generalization of Tukey reducibility that works well in $\le_D$1, yet under $\le_D$2 it collapses back to the classical relation (Sakai et al., 9 Jul 2025).

The principal open problem attached directly to the notion is whether the converse to that collapse holds: does the coincidence of pre-Tukey and ordinary Tukey reducibility imply $\le_D$3? This question isolates the set-theoretic strength of the multivalued completion principle built into $\le_D$4 (Sakai et al., 9 Jul 2025).

Recent work in adjacent directions has also isolated stronger witness notions inside ordinary Tukey theory. In particular, the “strong part of the Tukey spectrum” $\le_D$5 for a set $\le_D$6 of regular cardinals consists of those regular $\le_D$7 for which there is a witness $\le_D$8 of size $\le_D$9 such that every $E^*$00-sized subfamily has unboundedly many unbounded coordinates. This is not a reducibility relation, but it strengthens ordinary spectrum membership and can be used in place of PCF-theoretic scales to lift the existence of Jónsson algebras from below a singular cardinal to its successor (Gilton, 2022). This suggests a broader pattern: current research develops Tukey theory in two complementary ways, by weakening it to a choice-free relation such as $E^*$01, and by strengthening its witnesses in settings where additional structure is needed.