[#P2934] Effectivity of cocongruences in commutative monoids

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?

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1Context

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.

2Definitions

Definition 1 (Here p,q form a cocongruence when [p,q]:X direct-sum X ->Y). 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 2 (X amalgam_U X). 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.

3What 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.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (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.^[1]

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For cmon cocongruences effective, pins the dated research frontier: 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. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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] Relevance to this problem For Effectivity of cocongruences in commutative monoids, record cmon-cocongruences-effective-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake 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.

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 1. 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.

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ProblemEffectivity of cocongruences in commutative monoids

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Resolve the stated acceptance condition addresses Current status and unresolved remainder

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“Effectivity of cocongruences in commutative monoids.” TheoremDB. P2934. Problem statement; statement text SHA-256 176a947075d9fb202933d0111c8a0d4d3639c01b76e15bddf1584814da92a997. https://theoremdb.org/statement/?ref=P2934

BibTeX

@misc{theoremdb-problem-176a947075d9fb202933d0111c8a0d4d3639c01b76e15bddf1584814da92a997,
  title = {{Effectivity of cocongruences in commutative monoids}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 176a947075d9fb202933d0111c8a0d4d3639c01b76e15bddf1584814da92a997},
  url = {https://theoremdb.org/statement/?ref=P2934}
}

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This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net^[1], current as of July 31, 2026.

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1References

Packet source. 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. ↗forum · discovery source · checked 2026-07-31Source use: original summary.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.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.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.
MathOverflow: Is every cocongruence in CMon effective?, source checked for the TheoremDB status review (2026-07-31). Status evidence identified in the source record and checked at the linked publication. ↗website · primary source · checked 2026-07-31Source use: original summary.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.Also cited at definition, related properties, and commutative-monoid example entry.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.

An original CC0 textbook reformulation motivated by the cited MathOverflow category-theory question; all categorical conventions are restated.

[#P2934] Effectivity of cocongruences in commutative monoids

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