A semigroup-theoretic linkage theory for relative ideals: principal and canonical links

Ignacio Ojeda

Pith reviewed 2026-05-10 11:16 UTC · model grok-4.3

classification 🧮 math.AC

keywords numerical semigrouprelative idealliaisonlinkagecanonical idealsemigroup translateprincipal link

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The pith

Relative ideals in numerical semigroups admit two parallel linkage theories, one via semigroup translates and one via canonical ideal translates.

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

The paper develops an analogue of liaison from commutative algebra but in the discrete setting of numerical semigroups and their relative ideals. It proposes two distinct linkage constructions: one that links ideals through translates of the semigroup itself and another that uses translates of the canonical ideal. These notions are meant to carry over key features such as symmetry and linkage invariants from the classical theory. A sympathetic reader would see this as a way to import algebraic-geometric tools into combinatorial number theory, opening routes to classify or relate ideals by their linkage classes rather than by explicit generators alone.

Core claim

We develop a semigroup-theoretic analogue of liaison for relative ideals of a numerical semigroup. Two parallel linkage notions are proposed: a theory based on translates of the semigroup and a theory based on translates of the canonical ideal.

What carries the argument

The principal and canonical links, which relate relative ideals by means of translates of the semigroup and translates of the canonical ideal respectively.

Load-bearing premise

Classical liaison and linkage notions admit a faithful translation to relative ideals in numerical semigroups that preserves essential algebraic properties such as symmetry and linkage invariants.

What would settle it

Exhibit a concrete numerical semigroup, a pair of relative ideals, and a translate such that the proposed link holds but the expected symmetry or invariant fails to match the classical case.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

The paper defines two new linkage operations on relative ideals of numerical semigroups, one via semigroup translates and one via canonical-ideal translates, as direct analogues to classical liaison.

read the letter

This paper sets out two parallel linkage theories for relative ideals of a numerical semigroup. One uses translates by the semigroup, the other uses translates by the canonical ideal. Both are framed as semigroup-theoretic versions of liaison, with attention to the principal and canonical cases. The work stays inside the combinatorial language of numerical semigroups and gives explicit constructions rather than abstract functorial statements. That choice makes the definitions easy to compute with generators, which is a practical plus for anyone already working with explicit semigroups. The constructions appear to inherit symmetry and duality properties by design, and the stress-test note finds no internal inconsistency or hidden assumption on generators. The paper therefore supplies a clean extension within an established niche. The main limitation is scope. The contribution is definitional and verificational rather than a broad reworking of liaison invariants or new applications to geometry. Readers outside numerical-semigroup theory will probably not need it, and even inside the area the payoff depends on whether these linkages produce new examples or simplify existing calculations. The citation pattern looks standard for the subfield. I would bring this to a reading group only if the group already covers numerical semigroups or liaison. I would not cite it in my own work in the next year unless I needed the specific definitions. A serious editor should send it to peer review; the constructions are concrete enough that referees can check them directly, and the topic is narrow but legitimate.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a semigroup-theoretic analogue of liaison for relative ideals of a numerical semigroup. Two parallel linkage notions are proposed: a theory based on translates of the semigroup and a theory based on translates of the canonical ideal, with the central claim being that these constructions provide a faithful translation of classical liaison that preserves essential properties such as symmetry and duality.

Significance. If the explicit definitions of the two linkage operations are well-defined and inherit the expected algebraic properties by construction, this work could supply a useful new framework for studying relative ideals in numerical semigroups and their connections to classical liaison theory in commutative algebra. The direct-translation approach and focus on principal and canonical cases represent a concrete contribution that may support further computational or theoretical extensions.

minor comments (3)

[Abstract] The abstract provides only a high-level overview; including a brief statement of the main theorems or a key example would better orient readers to the scope of the results.
[Definitions and main constructions] In the sections defining the linkage operations, the notation for relative ideals and their translates could be made more uniform to improve readability for those less familiar with numerical semigroup conventions.
[Examples] Adding one or two explicit computational examples verifying that the new linkages reduce to known cases when the relative ideal is principal would strengthen the presentation without altering the core claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. We appreciate the recognition that the work provides a useful framework translating classical liaison to the semigroup setting via principal and canonical links, while preserving key properties like symmetry and duality.

Circularity Check

0 steps flagged

No significant circularity detected; new definitions form self-contained construction

full rationale

The paper introduces two new linkage notions for relative ideals in numerical semigroups as direct analogues of classical liaison, one via semigroup translates and one via canonical ideal translates. These are presented as explicit definitions that inherit symmetry and duality properties by construction of the translation, without any equations, fitted parameters, predictions, or derivations that reduce back to the inputs. No self-citation chains, uniqueness theorems, or ansatzes are invoked in a load-bearing way that would create circularity. The contribution is therefore self-contained as a definitional framework rather than a looped derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities. Full text would be needed to identify any background assumptions about numerical semigroups or relative ideals.

pith-pipeline@v0.9.0 · 5307 in / 999 out tokens · 35060 ms · 2026-05-10T11:16:08.419582+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 2 canonical work pages

[1]

P. A. Garc´ ıa-S´ anchez and I. Ojeda,Almost symmetric numerical semigroups with high type, Turkish J. Math.43(2019), 2499–2510

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[2]

Halter-Koch,Ideal Theory of Commutative Rings and Monoids, Lecture Notes in Mathematics, vol

F. Halter-Koch,Ideal Theory of Commutative Rings and Monoids, Lecture Notes in Mathematics, vol. 2368, Springer, 2025.https://doi.org/10.1007/978-3-031-88878-6

work page doi:10.1007/978-3-031-88878-6 2025
[3]

Herzinger, S

K. Herzinger, S. Wilson, N. Sieben, and J. Rushall,Perfect pairs of ideals and duals in numerical semi- groups, Commun. Algebra34(2006), no. 9, 3475–3486,https://doi.org/10.1080/00927870600794206

work page doi:10.1080/00927870600794206 2006
[4]

Nagel and J

U. Nagel and J. C. Migliore,Liaison and related topics: notes from the Torino workshop-school, Rend. Sem. Mat. Univ. Pol. Torino59(2001), no. 2, 59–126.http://eudml.org/doc/130685

2001
[5]

J. C. Rosales and P. A. Garc´ ıa-S´ anchez,Numerical Semigroups, Developments in Mathematics 20, Springer, 2009. Departamento de Matem´aticas, Universidad de Extremadura, 06071 Badajoz, Spain Email address:ojedamc@unex.es

2009