Gamified Katětov Order: Combinatorics & Computability

• Gamified Katětov Order is a game-theoretic variant of classical Katětov reducibility that adapts finite-query strategies over upper sets and filters.
• It reinterprets reducibility as an interactive finite-query game whose closure under well-founded Fubini iterations bridges combinatorial and categorical (Lawvere–Tierney) structures.
• The order unifies filter combinatorics and computability theory by providing robust invariants and highlighting coarser equivalences than traditional Rudin–Keisler reductions.

The gamified Katětov order is a game-theoretic variant of Katětov reducibility on upper sets over $\omega$, and dually on filters and ideals, introduced to make precise a connection that had long been only informal: the connection between combinatorial reducibility on subsets of $\omega$ and the Lawvere–Tierney order on local operators in the Effective Topos. Its defining feature is that it behaves like the classical Katětov order after closure under finite and well-founded Fubini iteration. In its computable form it is equivalent, and in the extended setting isomorphic, to the $\leq_{\mathrm{LT}}$-order, so that a single preorder simultaneously organizes aspects of filter combinatorics, realizability, and computability-theoretic complexity (Kihara et al., 8 Feb 2026, Kihara et al., 13 May 2026).

1. Classical background and motivation

The classical Katětov order is a preorder used to compare the combinatorial strength of ideals, filters, and upper sets. For upper sets $\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$, one formulation is

$$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$

equivalently, in the direct-image form used in the literature,

$$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$

The Rudin–Keisler order is stronger, requiring bi-implication under pullback rather than one-way reducibility (Kihara et al., 8 Feb 2026).

This background matters because the classical Katětov order is already structurally intricate. Among tall Borel ideals there is no minimal element in the Katětov order (Grebík et al., 2017), and even the interval between the Borel ideals $\mathcal{ED}$ and $\mathrm{Fin}\otimes\mathrm{Fin}$ contains a copy of $\mathcal P(\omega)/\mathrm{Fin}$, with increasing and decreasing chains of length $\mathfrak b$ and antichains of size $\omega$0 (Das et al., 2020). In forcing theory, Katětov reductions also control properties of co-ideal variants of Laver forcing, including the addition of Cohen reals and the Laver property (Rosa et al., 14 Jan 2026). The gamified order was introduced against this background, but with a different aim: to isolate the fragment of Katětov-type reducibility that matches the local-operator structure of the Effective Topos.

On the categorical side, an LT topology in $\omega$1 is an endomorphism $\omega$2 satisfying the usual monotonicity, inflationary, and idempotence axioms, and the Lawvere–Tierney order is

$\omega$3

This order effectively embeds the Turing degrees. The motivating problem was to understand what combinatorial mechanism controls this order, and the answer supplied by the gamified Katětov order is that the relevant mechanism is not one-shot Katětov reducibility but a game-theoretic closure of it under iterated Fubini operations (Kihara et al., 13 May 2026).

2. Definition and game-theoretic content

The gamified Katětov order reframes reducibility as a finite-query game. The classical Katětov condition can be read as a one-query interaction: Player I chooses a target large set $\omega$4, Player II chooses a witness set in $\omega$5, Player I selects an input point, and Player II applies a fixed map $\omega$6 in an attempt to land inside $\omega$7. The gamified version permits finitely many adaptive queries, encoded by a partial continuous strategy on finite sequences rather than by a single global map (Kihara et al., 8 Feb 2026).

A tree $\omega$8 is $\omega$9-branching when every node has $\leq_{\mathrm{LT}}$0-large immediate successor set. The formal reduction $\leq_{\mathrm{LT}}$1 is then defined by the existence of a partial continuous witness $\leq_{\mathrm{LT}}$2 such that for every $\leq_{\mathrm{LT}}$3 there is a $\leq_{\mathrm{LT}}$4-branching tree $\leq_{\mathrm{LT}}$5 with

$\leq_{\mathrm{LT}}$6

This turns the witness from a single map into a tree-guided strategy that can adapt to finitely many stages of information (Kihara et al., 8 Feb 2026).

The point of the game-theoretic reformulation is not merely presentational. The defining theorem identifies the formal preorder with a genuine finite-query Katětov game: Player II has a winning strategy in that game if and only if the gamified Katětov reduction holds. This explains the term “gamified” and also clarifies why the order is naturally interpreted as a reduction notion stable under controlled, adaptive use of information rather than only under a single pullback map.

3. Closure under well-founded Fubini powers

The central structural theorem states that the gamified Katětov order is equivalent to the classical Katětov order closed under well-founded iterations of Fubini powers (Kihara et al., 8 Feb 2026). This description is the conceptual core of the subject. It means that one starts from ordinary Katětov reducibility and then closes it under iterated Fubini products indexed by well-founded trees, so that reductions are stable under a recursive hierarchy of product-like constructions.

This viewpoint explains why the order is simultaneously coarser and more robust than the classical Katětov order. One-shot distinctions can disappear after sufficiently many well-founded Fubini iterations, but the resulting preorder still remembers nontrivial complexity. The survey formulation is that the gamified order is “essentially the usual Katětov order now closed under well-founded iterations of Fubini powers,” and that this closure is precisely what makes the order capable of matching the Effective Topos picture (Kihara et al., 13 May 2026).

A further refinement is the passage from upper sets to upper sequences, that is, countable sequences of upper sets. This extension mirrors a categorical passage on the topos side: the move from basic topologies to recursive joins of basic topologies. The computable version of the extended gamified Katětov order on upper sequences is isomorphic to the original $\leq_{\mathrm{LT}}$7-order, which indicates that well-founded Fubini iteration is not an ancillary technicality but the mechanism that converts a one-step combinatorial reduction into an order compatible with the recursive structure of $\leq_{\mathrm{LT}}$8 (Kihara et al., 13 May 2026).

4. Identification with the Lawvere–Tierney order

The decisive theorem of the subject is that the computable gamified Katětov order is equivalent to the $\leq_{\mathrm{LT}}$9-order on upper sets over $\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$0, and that the extended computable order on upper sequences is isomorphic to the full Lawvere–Tierney order on LT topologies in the Effective Topos (Kihara et al., 8 Feb 2026, Kihara et al., 13 May 2026). This gives a concrete combinatorial representation of an order that had been studied primarily in categorical and realizability-theoretic terms.

The identification is significant for two reasons. First, it shows that the LT order is tightly controlled by the combinatorics of filters and upper sets on $\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$1. Second, it gives the gamified order a status different from that of a purely set-theoretic refinement: it becomes the exact combinatorial analogue of a central categorical order. The survey emphasizes this as a bridge between a natural variant of Katětov order, a concrete representation of abstract topologies, and a hierarchy of computability notions referred to there as “computability by majority” (Kihara et al., 13 May 2026).

There is also a structural explanation for why filters and upper sets appear at all. Lee and van Oosten’s theorem, recalled in the survey, states that every LT topology in $\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$2 is a recursive join of basic topologies. This pushes the analysis toward basic topologies determined by families of subsets of $\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$3, especially upper sets, and thereby toward a reducibility notion on those families. The gamified Katětov order is the form that reducibility takes once one incorporates the recursive-join structure rather than restricting attention to single-step basic data.

5. Coarseness, richness, and non-linearity

The gamified Katětov order is strictly coarser than Rudin–Keisler and also strictly coarser than Katětov, in ZF (Kihara et al., 8 Feb 2026). Dually on ideals, this coarsening has a dramatic consequence: all MAD families collapse to a single equivalence class. In the classical Katětov order, MAD families support many distinct classes and large antichains; in the gamified order, those distinctions disappear.

This coarseness does not imply degeneracy. The original paper proves an infinite strictly ascending chain of ideal classes,

$\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$4

and also shows that the gamified Katětov order and Tukey order are incomparable on filters over $\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$5 (Kihara et al., 8 Feb 2026). The order therefore measures something different from cofinality type in the Tukey sense, and something more stable under iteration than the classical Katětov order.

A further sharpening is supplied by the non-linearity theorem: the gamified Katětov order embeds $\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$6, and hence contains a chain of length $\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$7 and an antichain of size $\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$8 (Kihara et al., 20 May 2026). The embedding is realized by assigning to each $\mathcal U,\mathcal V\subseteq\mathcal P(\omega)$9 a summable ideal $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$0, built from a partition $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$1 of $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$2 into finite intervals and a decreasing sequence of positive rationals $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$3, in such a way that

$$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$4

For appropriate infinite and co-infinite $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$5, one also has

$$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$6

where $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$7 is the asymptotic zero-density ideal (Kihara et al., 20 May 2026).

The proof technology reflects the mixed rigidity and flexibility of the preorder. Separation arguments in the original development use a new labeling and critical-node method on the canonical witness tree, while the non-linearity paper supplements this with pigeonhole arguments, interval decompositions, and canonization theorems from Ramsey theory, including the Canonical Ramsey Theorem and the Canonical Hindman Theorem (Kihara et al., 8 Feb 2026, Kihara et al., 20 May 2026).

6. Degree profiles and computability-theoretic consequences

The gamified/LT perspective yields a degree-spectrum invariant for filters. For a filter $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$8, the associated profile $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ (h^{-1}[A]\in\mathcal V),$$9 is the set of Turing degrees lying below $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$0 in the LT/gamified sense. This is always an initial segment of the Turing degrees, and in fact is stated to be a proper initial segment (Kihara et al., 8 Feb 2026).

The resulting degree theory is neither trivial nor universal. For every Turing degree $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$1 there exists a summable ideal $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$2 such that $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$3, so every Turing degree occurs in the profile of some filter. At the same time, there is no filter whose profile contains all Turing degrees. The profile therefore functions as a genuine invariant of filter complexity rather than as a vacuous coding device (Kihara et al., 8 Feb 2026).

For definable filters, the picture becomes sharply uniform. If $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$4 is a non-principal $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$5 filter, then

$$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$6

equivalently, for every $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$7,

$$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$8

This substantially generalises earlier results of van Oosten and Kihara, and shows that the gamified Katětov order is not merely a combinatorial gadget: it detects a robust logical threshold at the hyperarithmetic degrees (Kihara et al., 8 Feb 2026).

7. Place within Katětov theory and current outlook

The gamified Katětov order is best understood as a new robustness notion inside the broader universe of Katětov-style reducibilities. Classical Katětov order remains central in descriptive set theory, the structure theory of Borel ideals, and the analysis of co-ideal forcings; for example, it governs when co-ideal Laver forcing adds Cohen reals and when it has the Laver property (Rosa et al., 14 Jan 2026). The gamified order does not replace these uses. Instead, it isolates the part of Katětov reducibility that survives finite adaptive querying and well-founded Fubini iteration.

This placement also clarifies a common misunderstanding. Because the gamified order collapses all MAD families, it can appear at first sight to be too coarse to sustain serious structure. The available theorems contradict that interpretation: the infinite strict chain, the incomparability with Tukey order, and the embedding of $$\mathcal U \leq_{\mathrm K} \mathcal V \iff \exists h:\omega\to\omega\ \forall A\in\mathcal U\ \exists B\in\mathcal V\ (h[B]\subseteq A).$$9 together show that the order is highly non-linear and internally rich. Its coarseness is selective rather than indiscriminate.

The current research programme treats this as evidence that different notions of complexity in logic and topology are controlled by a common mechanism once they are placed in the right topological framework (Kihara et al., 13 May 2026). The survey highlights open questions about how the coarseness of the gamified Katětov order relative to classical filter orders compares with the coarseness inherent in the lattice of subtoposes of $\mathcal{ED}$0, which is order-reversingly bijective with the Lawvere–Tierney order. A plausible implication is that future progress will require simultaneous use of combinatorial set theory, descriptive set theory, computability theory, and categorical logic rather than any one of these viewpoints in isolation.