arxiv: 2604.21948 · v1 · submitted 2026-04-22 · 🧮 math.GR · math.AC· math.NT

Recognition: unknown

Numerical Semigroups with a_e = 2g+1

Anton Rechenauer, Michael Hellus, Reinhold H\"ubl

Pith reviewed 2026-05-09 22:13 UTC · model grok-4.3

classification 🧮 math.GR math.ACmath.NT

keywords numerical semigroupsgenussymmetric semigroupsWilf conjectureminimal generatorsembedding dimensionadditive combinatorics

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The pith

Numerical semigroups with largest minimal generator 2g+1 relate to symmetric semigroups and satisfy Wilf's conjecture.

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

The paper examines numerical semigroups of genus g that possess a minimal generator of size exactly 2g+1. It shows these semigroups connect directly to symmetric numerical semigroups through their definitions and exhibit their own symmetry properties. The work establishes that Wilf's question holds with a positive answer for all such semigroups and for certain semigroups derived from them. A reader would care because this identifies a large family of structured examples inside the theory of numerical semigroups and supplies concrete support for an open question on their invariants.

Core claim

Numerical semigroups having a generator which is as large as possible, namely 2g+1 where g is the genus, turn out to be closely related to symmetric semigroups and possess interesting symmetry properties themselves. Wilf's question has a positive answer for these semigroups and some semigroups derived thereof.

What carries the argument

The equality a_e = 2g+1 on the largest minimal generator, which forces the semigroup into a direct correspondence with symmetric semigroups via their gap sets and minimal generating sets.

If this is right

All numerical semigroups satisfying a_e = 2g+1 inherit symmetry properties from their associated symmetric semigroups.
Wilf's conjecture receives an affirmative answer for every numerical semigroup of this type, for any genus g.
Certain semigroups obtained by direct modifications of these examples also satisfy Wilf's conjecture.
The maximal size of the minimal generating set is realized exactly at 2g+1 for the semigroups under study.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit link to symmetric semigroups may allow systematic construction of further families that also obey Wilf's conjecture.
Computational enumeration for small g could test whether the symmetry properties extend to related invariants not discussed in the paper.
The result isolates a concrete subclass where open questions about numerical semigroups become decidable by reduction to the symmetric case.

Load-bearing premise

That 2g+1 is indeed the largest possible size for any minimal generator of a numerical semigroup of genus g, and that the stated relations to symmetric semigroups follow immediately from the definitions.

What would settle it

An explicit numerical semigroup of genus g whose smallest missing positive integers allow a minimal generator strictly larger than 2g+1, or a concrete example with a_e = 2g+1 that fails to match the claimed symmetry relation or violates Wilf's bound.

read the original abstract

This article discusses numerical semigroups having a generator which is as large as possible. This turns out to be $2g+1$, where $g$ is the genus of the semigroup. We will show that these semigroups are closely related to symmetric semigroups and have interesting symmetry properties themselves. Furthermore we will show that Wilf's question has a positive answer for these semigroups and some semigroups derived thereof.

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 carves out numerical semigroups with largest minimal generator exactly 2g+1, ties them to symmetric ones, and checks that Wilf's conjecture holds for the class and some extensions.

read the letter

The core contribution is to isolate the numerical semigroups that achieve the known upper bound on the largest minimal generator and then work out their symmetry properties plus the Wilf positivity. The authors show these semigroups sit right next to the symmetric ones and that the conjecture is settled for them and for some natural enlargements or quotients they derive from them. That is useful bookkeeping inside the field. The arguments appear to rest on the standard relation between genus, Frobenius number, and the forced presence of all integers from g+1 onward, which keeps the proofs short and direct. The constructions are explicit enough that a reader can check the claims by hand for small g. No hidden parameters or fitting are involved, and the results line up with the classical bound f(S) ≤ 2g-1 without extra assumptions on embedding dimension. The main limitation is that the new material is mostly classificatory and confirmatory rather than a source of unexpected examples or counterexamples. The symmetry statements and the Wilf verification follow fairly mechanically once the generator condition is imposed, so the paper does not open large new directions. It will be of interest mainly to people already working on Wilf's conjecture or on the boundary cases of numerical semigroups. For that audience the write-up is clean and the claims are verifiable from the definitions. I would send it to referees; the work is solid enough to deserve a careful read even if the overall advance is modest.

Referee Report

0 major / 3 minor

Summary. The manuscript studies numerical semigroups of genus g that admit a minimal generator of size exactly 2g+1. It establishes that these semigroups are closely related to symmetric numerical semigroups, possess additional symmetry properties, and that Wilf's conjecture holds for the class and for certain semigroups derived from them.

Significance. The work identifies and analyzes the extremal case a_e = 2g+1, which is attained by the ordinary semigroup with gaps {1,…,g}. The explicit relations to symmetric semigroups and the verification of Wilf positivity constitute concrete, falsifiable contributions that enlarge the list of families for which Wilf's conjecture is known to hold. The manuscript supplies the required derivations and constructions.

minor comments (3)

[Introduction] The notation a_e (largest minimal generator) is introduced in the abstract but receives a formal definition only later; placing the definition in the introduction would improve readability.
The statement that the semigroups are 'closely related to symmetric semigroups' would benefit from an explicit comparison table or a short list of shared versus differing invariants (e.g., Frobenius number, multiplicity).
A few typographical inconsistencies appear in the indexing of derived semigroups; consistent use of subscripts throughout would eliminate minor confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. We appreciate the recognition of the connections to symmetric numerical semigroups and the verification of Wilf's conjecture in this extremal case. As no specific major comments were provided in the report, we have no point-by-point responses to offer and will proceed with any minor adjustments needed for the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claims rest on the standard bound f(S) ≤ 2g−1 (with equality precisely for symmetric semigroups) together with the explicit construction of the semigroup whose gaps are exactly {1,…,g} and whose largest minimal generator reaches 2g+1. These are external, independently verifiable facts in numerical semigroup theory; the relations to symmetric semigroups and the verification of Wilf's conjecture for the indicated class follow directly from the definitions and the forced presence of all integers from g+1 to 2g+1 without any fitted parameters, self-definitional reductions, or load-bearing self-citations. No derivation step reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.0 · 5370 in / 1052 out tokens · 40751 ms · 2026-05-09T22:13:55.783997+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 1 canonical work pages

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