# P2934: Effectivity of cocongruences in commutative monoids

- ID: `P2934`
- Reference: `cmon-cocongruences-effective`
- Page: https://theoremdb.org/statements/P2934
- Record maturity: Reviewed problem with recorded work

## Problem

In the category \(\mathbf{CMon}\) of commutative monoids, is every cocongruence effective? Concretely, if \(p,q:X\rightrightarrows Y\) form a cocongruence and \(U=\{x\in X:p(x)=q(x)\}\), must the canonical map \(X\amalg_U X\to Y\), \([x,x']\mapsto p(x)+q(x')\), be an isomorphism?

### Problem setup

- **Remark.** Here p,q form a cocongruence when [p,q]:X direct-sum X ->Y is an epimorphism and, for every commutative monoid T, the image of Hom(Y,T)->Hom(X,T)^2 is an equivalence relation.
- **Definition.** X amalgam_U X is the pushout of the two inclusions U->X in commutative monoids, and the displayed canonical map is induced by p on the first copy and q on the second.

### What counts as a solution

- Prove that the canonical map X amalgam_U X ->Y is an isomorphism for every cocongruence p,q, or give explicit finitely presented commutative monoids and maps forming a cocongruence for which the map fails to be an isomorphism.
- A counterexample must include checkable proofs of the epimorphism and Hom-equivalence-relation conditions, plus a witness to failure of injectivity or surjectivity of the canonical map.

## Status

UNKNOWN as of 2026-07-31. A recent answer proves effectivity when X admits a positive degree map to the natural numbers, including free commutative monoids. Its comments continue to clarify details, and the unrestricted case remains unresolved on the page. Prove that the canonical map X amalgam_U X ->Y is an isomorphism for every cocongruence p,q, or give explicit finitely presented commutative monoids and maps forming a cocongruence for which the map fails to be an isomorphism. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** UNKNOWN as of 2026-07-31. A recent answer proves effectivity when X admits a positive degree map to the natural numbers, including free commutative monoids. Its comments continue to clarify details, and the unrestricted case remains unresolved on the page. Prove that the canonical map X amalgam_U X ->Y is an isomorphism for every cocongruence p,q, or give explicit finitely presented commutative monoids and maps forming a cocongruence for which the map fails to be an isomorphism.

UNKNOWN as of 2026-07-31. A recent answer proves effectivity when X admits a positive degree map to the natural numbers, including free commutative monoids. Its comments continue to clarify details, and the unrestricted case remains unresolved on the page.

A complete resolution must satisfy this condition: Prove that the canonical map X amalgam_U X ->Y is an isomorphism for every cocongruence p,q, or give explicit finitely presented commutative monoids and maps forming a cocongruence for which the map fails to be an isomorphism.

### Background and intake notes

Finite presentations, pushout normal forms, and degree filtrations from partial cases can be reused. The Hom condition is global in the test monoid, so finite counterexample searches need a justified reduction to a bounded family of tests.

- Original intake status: UNKNOWN as of 2026-07-27. A recent answer proves effectivity when X admits a positive degree map to the natural numbers, including free commutative monoids. Its comments continue to clarify details, and the unrestricted case remains unresolved on the page.
- The MathOverflow question, its answer, and every visible comment were checked on 2026-07-27. The answer handles a degree-positive class and does not claim the theorem for all commutative monoids.
- The category-property database entry on coquotients of cocongruences was checked for terminology and neighboring categorical facts. It did not provide a proof that CMon has the requested property.
- Searches for effective equivalence relations in the opposite of CMon, corelations of commutative monoids, and exactness properties found broader categorical results without an exact resolution of the displayed map.
- A local corpus search for CMon cocongruence, effective cocongruence, and pushout equalizer found no duplicate.

- Recorded example: The source answer proves the claim when X has a degree map X->N whose zero fiber is {0}; this includes finitely generated free commutative monoids.

### Open directions

- **Route 1** (reported): Prove that the canonical map X amalgam_U X ->Y is an isomorphism for every cocongruence p,q, or give explicit finitely presented commutative monoids and maps forming a cocongruence for which the map fails to be an isomorphism. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `cmon-cocongruences-effective`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>MathOverflow: Is every cocongruence in CMon effective?. Question 511792, its partial answer, and every visible comment were checked on 2026-07-27. Question 511792, its partial answer, and every visible comment were checked on 2026-07-27. https://mathoverflow.net/questions/511792/is-every-cocongruence-in-mathbfcmon-effective
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For Effectivity of cocongruences in commutative monoids: UNKNOWN as of 2026-07-27. A recent answer proves effectivity when X admits a positive degree map to the natural numbers, including free commutative monoids. Its comments continue to clarify details, and the unrestricted case remains unresolved on the page.
   - Source named by the research packet.
2. <a id="reference-2"></a>CatDat, “coquotients of cocongruences,” category-property database entry, checked 2026-08-01. definition, related properties, and commutative-monoid example entry https://catdat.app/category-property/coquotients_of_cocongruences
   - website; reference source; checked 2026-07-31
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Effectivity of cocongruences in commutative monoids, this source fixes the categorical terminology used by the target; the unrestricted status comes from the cited MathOverflow discussion.