A poset game on submonoids of additively indecomposable ordinals

TL;DR

This work studies a Chomp-style poset game on submonoids of additively indecomposable ordinals under natural sum and product, focusing on posets generated by finite sets of ordinals (wpog) and the resulting well-partial order structure. It establishes a Hanf-number-type phenomenon: for each fixed generator class there exists a least ordinal $\xi$ such that either the second player wins on all larger levels or only on a tail up to $\alpha\le\xi$, with the first player winning thereafter; the bound is refined to $\xi<|\omega^\sigma|^+$ when the generator set is finite. The paper also proves that for finite generator sets, $\text{ch}(\Gamma)<\omega_1$ if $\Gamma\subset\omega$, and $\text{ch}(\Gamma)<|\max\Gamma|^+$ otherwise, using definability and embedding arguments, and it provides explicit winning strategies in a particular family. Overall, the work blends set-theoretic methods with combinatorial game theory to bound and construct strategies for Chomp on ordinal-based monoids, offering a framework for understanding strategy existence via Hanf-number-like thresholds.

Abstract

Inspired by García-Marco and Knauer arXiv:1705.11034, we consider the Chomp game on a natural well-partial order structure defined on submonoids $\mathcal{S}^σ\subseteqω^σ$ of additively indecomposable ordinals generated by sets of ordinals under natural sum and product of ordinals. We have called the generatinng sets of ordinals for which the game terminates in a finite number of steps well-partial order generators, and, under very mild conditions, we have characterized those that are finite. A fundamental observation is that, for every ordinal $σ$, there exists an ordinal $ξ$ such that, for any class $\{ (\mathcal{S}^τ;\leq_{\mathcal{S}^τ}) \mid τ\geqσ\}$ of well-partial orders generated by a subset of $ω^σ$, the second player has a winning strategy on all those posets or only on $\{ (\mathcal{S}^β;\leq_{\mathcal{S}^β}) \midβ\inα\}$ for some successor ordinal $α\leqξ$ (and the first player will have a winning strategy on the rest of the posets). This Hanf number-style property could be valuable in proving the existence of winning strategies. We conjecture that $ ξ$ is the smallest possible and, using results from arXiv:1908.09664, we prove that, if we restrict ourselves to the classes generated by finite sets of ordinals, $ξ<|ω^σ|^+$. We also explicitly describe a winning strategy for a specific family of classes.

Figures (1)

• Figure 1: Posets $(\mathcal{S}^\sigma;\leq_{\mathcal{S^\sigma}})$ with $\mathcal{S}^\sigma=\langle3,5\rangle^\sigma$ and $\sigma\in\{1,2,3\}$.

Theorems & Definitions (41)

• Theorem 2.1: Cantor normal form for base $\omega$
• Definition 2.2: Natural sum and product
• Lemma 2.3
• Definition 2.5
• Definition 2.6: Ordinal subtraction
• Lemma 2.7: Poset structure
• proof
• Theorem 2.8: Zermelo's theorem
• Proposition 2.9
• proof