Combinatorial or game-theoretic proofs of the cancellation law for surjective cardinals

The paper proves that surjective cardinals form a surjective cardinal algebra and establishes a choice-free cancellation law: for all cardinals a, b and all nonzero natural numbers m, if m·a =* m·b then a =* b. This extends classical cancellation phenomena to the surjective-cardinal setting.

For the classical Bernstein division theorem, there are known combinatorial and game-theoretic proofs (due to Tarski and Schwartz respectively). The authors highlight the potential for analogous methods to handle surjective cardinals but note that Tarski’s combinatorial proof relies heavily on the refinement postulate for cardinals, suggesting that adapting such techniques may be challenging in the surjective-cardinal framework where only finite refinement is available.