Characterization of the game-equivalence lattice classes

Characterize the class of infinite lattices with greatest element 1 for which the Menger selection principle implies that Player I has no winning strategy in the corresponding Menger game, and the corresponding class for the Rothberger selection principle and game.

Background

The paper defines two classes of infinite lattices with greatest element 1. The class H\mathcal H consists of lattices for which the Menger selection principle (V1,V1)(\mathcal V_1,\mathcal V_1) implies that Player I has no winning strategy in the Menger game Gfin(V1,V1)\mathsf G_\mathrm{fin}(\mathcal V_1,\mathcal V_1). The class P\mathcal P is defined analogously for the Rothberger principle and game, (V1,V1)(\mathcal V_1,\mathcal V_1) and G1(V1,V1)\mathsf G_1(\mathcal V_1,\mathcal V_1).

Theorems 3.4 and 4.4 establish sufficient distributivity hypotheses for membership in these classes, but do not characterize the classes themselves. The question asks for necessary and sufficient structural conditions.

References

Characterize the classes $\mathcal H$ and $\mathcal P$.

— Primeless proofs of the Menger and Rothberger games  (2609.28780 - Mezabarba, 23 Sep 2026) in Question 7.1, Section 7 (Further comments and questions)

Can $C_1$ be replaced by a strictly weaker sufficient condition for $\mathcal H$? For instance, Example~\ref{ex:binary_pawlikowski} shows that $C_1$ alone does not suffice for $\mathcal P$. Is there a sufficient condition for $\mathcal P$ strictly weaker than $C_{\omega,1}$?

— Primeless proofs of the Menger and Rothberger games  (2609.28780 - Mezabarba, 23 Sep 2026) in Paragraph immediately following Question 7.1, Section 7 (Further comments and questions)

It remains to determine whether other ideal constructions admit complete lattice representations with a similar control of cardinality.

— Primeless proofs of the Menger and Rothberger games  (2609.28780 - Mezabarba, 23 Sep 2026) in Final paragraph of Section 7 (Further comments and questions)