Metadata
bibkey: jack2026conjugacy
authors: Trevor Jack
year: 2026
title: Answering Five Open Problems Involving Semigroup Conjugacy
doi: 10.1007/s00233-026-10614-w
url: https://arxiv.org/abs/2411.16284
claim: Section 2 defines Otto conjugacy and Section 9, Problem 9.4 asks whether a finite semigroup can have an Otto congruence with a noncancellative quotient.
strata_touched:
- D5/S3/FiniteGroups/FiniteOttoCongruence
license: citation-only
triage: anchor
Verified locator
DOI: https://doi.org/10.1007/s00233-026-10614-w
Source: https://arxiv.org/abs/2411.16284
Source statements
Semigroup Forum 112 (2026), 450–466. The quotations and page numbers below refer to arXiv:2411.16284v1.
Section 2, p. 2:
∀a, b ∈ S (a ∼o b ⇔ ag = gb and bh = ha for some g, h ∈ S).
Section 9, Problem 9.4, p. 14:
Is there a finite semigroup for which ∼o is a congruence and S/∼o is not cancelative?
The two witnesses belong to S, without adjoining an identity. Congruence means multiplication compatibility of this equivalence relation; cancellation of the quotient is expressed by left and right cancellation on its representatives. The source’s Section 6 constructs an infinite noncancellative quotient. The finite negative answer is derived in the associated Lean module.
The arXiv source has been read. The Semigroup Forum version of record and Liu–Wang, Communications in Algebra 53 (2025), 375–389 have not been read; completeness of the literature check is ASSUMED-UNVERIFIED.