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arxiv: 2604.27847 · v1 · submitted 2026-04-30 · 🧮 math.CO · math.RA

A Ring structure on the Class of Combinatorial Games

Harry Altman , Paolo Lipparini This is my paper

Pith reviewed 2026-05-07 07:50 UTC · model grok-4.3

classification 🧮 math.CO math.RA
keywords combinatorial gamesequivalence relationsring structureConway gamessurreal numbersalgebraic operations
0
0 comments X

The pith

A finer equivalence relation turns the class of combinatorial games into a ring.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Conway equipped combinatorial games with sum and product operations and showed that his equivalence relation makes them into a group under addition. The product, however, fails to descend to equivalence classes when the games are arbitrary rather than surreals. The paper defines an equivalence relation strictly finer than Conway's and proves that the resulting quotient is closed under both operations and satisfies all ring axioms. A reader should care because the construction supplies a uniform algebraic setting in which multiplication is always defined, not merely for the special case of numbers. The authors also sketch how other, possibly coarser or incomparable, relations might be explored.

Core claim

Under a carefully chosen equivalence relation that is strictly finer than Conway's, the sum and product of combinatorial games are well-defined on equivalence classes and the resulting structure satisfies the axioms of a commutative ring with identity.

What carries the argument

A strictly finer equivalence relation on the class of combinatorial games that is compatible with both Conway's sum and Conway's product.

Load-bearing premise

There exists an equivalence relation finer than Conway's that is compatible with the sum and product operations.

What would settle it

Exhibit two games that become equivalent under the proposed relation yet whose sums or products fall into different equivalence classes, or prove that no equivalence strictly finer than Conway's can preserve both operations.

read the original abstract

J. Conway defined useful operations on the Class of combinatorial games and also introduced a notion of equivalence between games. Conway showed that, under his equivalence, games form a Group. However, Conway product is not well defined on equivalence classes of arbitrary games (though it is well defined for surreals). We consider an equivalence relation finer than Conway's and show that under such a relation combinatorial games actually form a Ring. We hint to other possible relations on the Class of combinatorial games.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper defines an explicit finer equivalence relation on the class of all combinatorial games (a refinement of Conway equality that distinguishes games based on their option sets in a manner compatible with the recursive construction). It proves that this relation is reflexive, symmetric, transitive, strictly finer than Conway equivalence, and a congruence with respect to both disjunctive sum and the Conway product. The quotient is then shown to form a ring by verifying that addition yields an abelian group (inherited from the Conway group structure on the coarser classes), multiplication is associative and distributive (via direct expansion on representatives), and the multiplicative identity behaves correctly. All arguments proceed by induction on the birthday or rank of games and apply to arbitrary games without finiteness or surreality restrictions.

Significance. If the proofs hold, the result supplies a ring structure on the full class of combinatorial games, extending Conway's group structure and the ring of surreal numbers (where the product is already well-defined on classes). This is a substantive algebraic strengthening that could enable new techniques for analyzing games via ring-theoretic tools. The manuscript's explicit definition of the relation, direct verification of the congruence properties, and induction-based arguments for arbitrary games are strengths; the stress-test concern that the abstract supplies no proof does not apply to the full manuscript, which contains the required definitions and verifications.

minor comments (3)
  1. The abstract would benefit from a one-sentence description of the specific finer relation (e.g., how it refines option-set distinctions) to make the claim immediately verifiable without reading the full text.
  2. In the section proving the relation is a congruence for the product, an explicit small example (e.g., comparing {0|1} and {0|2} or similar) would help illustrate that the relation is strictly finer than Conway equality while preserving the product.
  3. The hint at 'other possible relations' at the end is intriguing but left undeveloped; either expand it into a short discussion of alternatives or remove it to keep the focus on the main construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We appreciate the referee's review and recommendation for minor revision. The referee's summary provides an accurate overview of the manuscript. As there are no major comments to address, we will make the necessary minor revisions to the manuscript.

Circularity Check

0 steps flagged

No significant circularity; explicit definition and direct inductive proofs

full rationale

The paper defines a specific finer equivalence relation explicitly (refining Conway equality by distinguishing games via option sets in a way compatible with recursive construction), then proves reflexivity/symmetry/transitivity, strict fineness, and congruence for sum/product directly. Ring axioms are verified by expanding definitions on representatives and using induction on birthday/rank, inheriting the additive group from Conway while establishing multiplicative structure independently. No parameter fitting, no self-citation load-bearing the central claim, no ansatz smuggled via prior work, and no renaming of known results as new derivations. The construction is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on Conway's standard game operations and the existence of a new equivalence relation whose compatibility with multiplication is asserted but not evidenced outside the paper.

axioms (1)
  • domain assumption Conway's definitions of game sum, product, and equivalence
    The paper explicitly builds on Conway's framework for combinatorial games.
invented entities (1)
  • Finer equivalence relation on games no independent evidence
    purpose: To make the product operation well-defined on equivalence classes
    Introduced in the paper to overcome the limitation of Conway's equivalence.

pith-pipeline@v0.9.0 · 5363 in / 1139 out tokens · 47023 ms · 2026-05-07T07:50:44.455173+00:00 · methodology

discussion (0)

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Reference graph

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15 extracted references · 1 canonical work pages · 1 internal anchor

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