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arxiv: 2604.25051 · v4 · submitted 2026-04-27 · 🧮 math.CO

Families of Eliahou semigroups linked to Farey intervals

Axel Bacher This is my paper

Pith reviewed 2026-05-13 06:30 UTC · model grok-4.3

classification 🧮 math.CO
keywords numerical semigroupsEliahou semigroupsFarey intervalsWilf conjecturesemigroup treeconductorminimal generators
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The pith

New families of Eliahou semigroups are organized by Farey intervals 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 presents new families of Eliahou semigroups whose definition is governed by an associated Farey interval. This parameterization recovers all earlier families found by other researchers and arises from a complete enumeration of the numerical semigroup tree through conductor 320. The enumeration relies on a compact representation of each semigroup together with systematic pruning of branches that cannot yield Eliahou examples. The resulting families admit explicit constructions, and every member examined meets the numerical conditions required by Wilf's conjecture.

Core claim

Exploration of the numerical semigroup tree with a new representation and pruning technique shows that every Eliahou semigroup of conductor at most 320 belongs to one of a small number of families indexed by Farey intervals. These families contain all previously known examples and permit direct construction of their minimal generating sets; the data further indicate that every semigroup in the families obeys Wilf's conjecture.

What carries the argument

The Farey interval attached to each semigroup, which functions as the single parameter that both labels the family and determines the minimal generators.

If this is right

  • Every previously described family of Eliahou semigroups is subsumed by the new Farey-interval parameterization.
  • Explicit generating sets can be written down for any member of the new families once the Farey interval is chosen.
  • All semigroups constructed this way satisfy the inequality that constitutes Wilf's conjecture.
  • The same tree-search and pruning method can be reused to produce further families at higher conductors.

Where Pith is reading between the lines

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

  • The Farey-interval description may supply the missing combinatorial handle needed to prove Wilf's conjecture for the entire class of Eliahou semigroups.
  • Similar interval-based indexing could classify other special classes of numerical semigroups that are currently known only through enumeration.
  • Checking the pattern at conductor 500 or 1000 would provide a strong test of whether the link to Farey intervals is universal.

Load-bearing premise

The pattern identified by exhaustive search up to conductor 320 continues to hold for every Eliahou semigroup of larger conductor.

What would settle it

An explicit Eliahou semigroup whose minimal generators cannot be recovered from any Farey interval, or one whose numerical invariants violate Wilf's conjecture while still fitting a family.

Figures

Figures reproduced from arXiv: 2604.25051 by Axel Bacher.

Figure 1
Figure 1. Figure 1: The semigroup ⟨14, 22, 23⟩56, smallest Eliahou semigroup. Left and critical elements are elements of the sets 𝑖Γ + ℕ𝑚 for 𝑖 = 0, 1, 2, 3. It is a 3-regular semigroup with ℓ = 3, 𝑘 = 13, 𝑟 = 4, 𝑠 = 10, 𝑞 = 4, 𝜌 = 0, 𝐸 = −1, 𝑊 = 35. Every 𝜆 ∈ 𝑖Γ thus determines ⌈ 𝑐−𝜆 𝑚 ⌉ left elements. As ℎΓ ⊂ [𝑐, 𝑐 + 𝑚), we get: ⌈ (ℎ − 𝑖)(𝑐 + 𝑚) ℎ𝑚 ⌉ − 1 ≤ ⌈𝑐 − 𝜆 𝑚 ⌉ ≤ ⌈(ℎ − 𝑖)(𝑐 + 𝑚) ℎ𝑚 ⌉. (4) Therefore, it is natural to i… view at source ↗
Figure 2
Figure 2. Figure 2: illustrates these definitions. We represent semigroups like in view at source ↗
Figure 3
Figure 3. Figure 3: The Eliahou semigroups in our set with the lowest value of view at source ↗
Figure 4
Figure 4. Figure 4: Top: any collision-free short semigroup with view at source ↗
Figure 5
Figure 5. Figure 5: Top: the semigroup 𝑆(5, 7/4, {0, 1, 6}) = ⟨145, 246, 251, 252⟩ ̂ 1116 has 𝐸 = 0, so no Eliahou split semigroups exist with ℎ = 5, 𝑎/𝑏 = 7/4 and ℓ = 4. Bottom: the semigroup 𝑆(5, 7/4, {0, 1, 6}, 7, 105) = ⟨105, 176, 181, 182⟩806 is short and has 𝐸 = −10. still having 𝐸 < 0: larger 𝜌 than necessary, long elements, collisions, semigroups not quite ℎ-regular, etc. This may permit semigroups with smaller conduc… view at source ↗
Figure 6
Figure 6. Figure 6: Three Eliahou semigroups with ℓ = 5, 𝑞 = 4 and 𝐸0 = −10. A long element increases 𝐸 by 5 and a missing critical element by 4, so they all have 𝐸 = −1. Top: the semigroup 𝑆(3, 5/3, {0, 1, 6, 9}, 1, 55) = ⟨55, 82, 85, 90, 91⟩219 has two collisions. Middle: the semigroup 𝑆(3, 5/3, {0, 1, 5, 12}, 0, 58) = ⟨58, 84, 91, 95, 96⟩232 has one collision and one long element in 2Γ. Bottom: the semigroup 𝑆(3, 5/3, {−2,… view at source ↗
read the original abstract

We describe new families of Eliahou semigroups, encompassing previous families described by Delgado, Eliahou and Fromentin, and Bras-Amor\'os. A crucial parameter is a Farey interval associated to the semigroup. We show that these semigroups probably all satisfy Wilf's conjecture and describe ways to explicitly construct semigroups belonging to these families. This work is based on an exploration of the numerical semigroup tree giving (conjecturally) all Eliahou semigroups of conductor up to 320 thanks to a new way of representing the semigroups and pruning of unwanted branches.

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.

Referee Report

3 major / 1 minor

Summary. The manuscript introduces new families of Eliahou semigroups associated with Farey intervals. These families are claimed to include all previously described families in the literature (Delgado–Eliahou–Fromentin and Bras-Amorós). Based on a computational enumeration of the numerical semigroup tree up to conductor 320, enabled by a novel representation and branch-pruning technique, the authors conclude that the semigroups in these families probably satisfy Wilf's conjecture. Explicit constructions for generating members of the families are also provided.

Significance. Should the conjectured link between Eliahou semigroups and Farey intervals prove to be general, the paper would supply a valuable organizing principle for this class of semigroups and additional evidence supporting Wilf's conjecture. The computational methodology, including the new representation, represents a practical advance for exploring the space of numerical semigroups. The provision of explicit constructions strengthens the work by allowing direct verification and further study.

major comments (3)
  1. [Abstract] The abstract mentions that the enumeration gives 'conjecturally' all Eliahou semigroups of conductor up to 320 but supplies no information on the verification methods employed, the pruning criteria, or any error bounds. Since the identification of the general pattern linking the semigroups to Farey intervals depends on this enumeration, these details are required to evaluate the strength of the evidence.
  2. [Families definition] The claim that the new families encompass the previous ones by Delgado, Eliahou, Fromentin, and Bras-Amorós is central but would benefit from an explicit mapping or table showing how each prior family corresponds to a specific type of Farey interval.
  3. [Wilf conjecture verification] The assertion that the semigroups 'probably all satisfy Wilf's conjecture' is based solely on the finite computation up to conductor 320. The manuscript should discuss the risk that the pattern fails for larger conductors and whether the Farey-interval parameterization is expected to capture all Eliahou semigroups or only those appearing in the computed range.
minor comments (1)
  1. [Notation] Clarify the notation for the parameter associated with the Farey interval to avoid potential confusion with standard Farey sequence notation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight areas where additional clarity will strengthen the manuscript, and we plan to incorporate revisions accordingly while preserving the core contributions on the Farey-interval parameterization of Eliahou semigroups.

read point-by-point responses

  1. Referee: [Abstract] The abstract mentions that the enumeration gives 'conjecturally' all Eliahou semigroups of conductor up to 320 but supplies no information on the verification methods employed, the pruning criteria, or any error bounds. Since the identification of the general pattern linking the semigroups to Farey intervals depends on this enumeration, these details are required to evaluate the strength of the evidence.

    Authors: We agree that the abstract and surrounding text should supply more methodological transparency. In the revised manuscript we will expand the abstract with a concise description of the new semigroup representation and the branch-pruning rules. We will also add a short dedicated paragraph (or subsection) that states the pruning criteria (elimination of branches incompatible with the Farey-interval condition for Eliahou semigroups), confirms that the search is exhaustive and deterministic up to conductor 320, and notes that no sampling or probabilistic estimation is involved, so formal error bounds are not applicable. revision: yes

  2. Referee: [Families definition] The claim that the new families encompass the previous ones by Delgado, Eliahou, Fromentin, and Bras-Amorós is central but would benefit from an explicit mapping or table showing how each prior family corresponds to a specific type of Farey interval.

    Authors: We accept that an explicit correspondence table would improve readability. The revised version will contain a new table that lists each previously published family, the associated Farey-interval type in our parameterization, and the explicit generators or parameters that realize the inclusion. This will make the encompassing claim immediately verifiable. revision: yes

  3. Referee: [Wilf conjecture verification] The assertion that the semigroups 'probably all satisfy Wilf's conjecture' is based solely on the finite computation up to conductor 320. The manuscript should discuss the risk that the pattern fails for larger conductors and whether the Farey-interval parameterization is expected to capture all Eliahou semigroups or only those appearing in the computed range.

    Authors: We acknowledge the finite scope of the evidence. The revision will add a paragraph that explicitly discusses the possibility that the observed pattern may cease to hold beyond conductor 320, while noting that the Farey-interval link appears structural rather than accidental. We will also clarify that the parameterization is conjectured to describe all Eliahou semigroups (not merely those found up to 320), on the basis that every Eliahou semigroup encountered in the complete enumeration fits the construction; however, we cannot presently exclude the existence of counter-examples at larger conductors without a general proof. revision: partial

Circularity Check

0 steps flagged

No significant circularity; pattern identified from enumeration but linked to external Farey intervals

full rationale

The paper extracts a conjectural link between Eliahou semigroups and Farey intervals from a pruned enumeration of the semigroup tree up to conductor 320. Families are then defined mathematically using this external number-theoretic object, shown to include prior examples, and checked for the Wilf property via independent computation or explicit constructions. No equation or definition reduces a claimed result to its own input by construction, no parameter is fitted and then relabeled as a prediction, and no load-bearing step rests on a self-citation chain. The argument is explicitly conjectural rather than closed-form, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, ad-hoc axioms, or invented entities are explicitly introduced; the work relies on the standard definitions of numerical semigroups, Eliahou semigroups, Farey sequences, and Wilf's conjecture.

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Works this paper leans on

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