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arxiv: 2604.26901 · v1 · submitted 2026-04-29 · 🧮 math.NT · math.CO· math.RA

On the automorphisms of the power semigroups of a numerical semigroup

Salvatore Tringali , Kerou Wen This is my paper

Pith reviewed 2026-05-07 10:25 UTC · model grok-4.3

classification 🧮 math.NT math.COmath.RA
keywords numerical semigroupspower semigroupsautomorphism groupssumset operationtrivial automorphismsadditive monoids
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The pith

The automorphism group of the power semigroup of any numerical semigroup is trivial.

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

A numerical semigroup is a cofinite additive submonoid of the non-negative integers. The power semigroup P(H) is the collection of all its non-empty subsets equipped with the sumset operation. The paper proves that the only automorphism of this semigroup is the identity map. The same conclusion holds for the submonoid P0(H) of subsets containing zero, whenever zero lies in H. This establishes that the sumset structure on subsets admits no non-trivial rearrangements that preserve the operation.

Core claim

If H is a numerical semigroup, then the automorphism group of P(H), the semigroup of its non-empty subsets under the sumset operation, is trivial. When 0 belongs to H, the automorphism group of the submonoid P0(H) of subsets containing 0 is likewise trivial. The proofs combine combinatorial counting arguments on subsets with basic properties of semigroups.

What carries the argument

The power semigroup P(H) of non-empty subsets of H under the sumset operation, whose automorphisms are shown to reduce exactly to the identity map.

If this is right

  • Any bijection of the non-empty subsets that preserves all sumsets must fix every subset.
  • The addition in H is encoded so rigidly in the sumset table that no other symmetry is possible.
  • The same rigidity applies directly to the monoid version P0(H) whenever 0 is present in H.
  • Combinatorial arguments on finite versus infinite subsets suffice to rule out non-trivial automorphisms.

Where Pith is reading between the lines

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

  • The result suggests that a numerical semigroup might be recoverable up to isomorphism from its power semigroup alone.
  • Similar triviality could be checked for power semigroups over other cofinite monoids in the integers.
  • The proof ideas might extend to show that certain sumset identities force the underlying set to be rigid.

Load-bearing premise

H is a cofinite additive submonoid of the non-negative integers.

What would settle it

Produce a numerical semigroup H and a bijection f on its non-empty subsets, other than the identity, such that f(A + B) = f(A) + f(B) holds for every pair of subsets A and B.

read the original abstract

If $H$ is a numerical semigroup (that is, a cofinite subset of the non-negative integers closed under addition), then the non-empty subsets of $H$ form a semigroup $\mathcal P(H)$ under the sumset operation induced by addition in $H$. Moreover, if $0 \in H$, then $\mathcal P(H)$ is a monoid with identity element $\{0\}$, and the family $\mathcal P_0(H)$ of all subsets of $H$ containing $0$ is a submonoid of $\mathcal P(H)$. We show that the automorphism group of $\mathcal P(H)$ is trivial, and the same holds for $\mathcal P_0(H)$ when $0 \in H$. The proofs blend ideas from combinatorics and semigroup theory.

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

0 major / 2 minor

Summary. The manuscript proves that if H is a numerical semigroup, then the automorphism group of the power semigroup P(H) under sumset is trivial. It further proves that the automorphism group of the submonoid P0(H) is trivial whenever 0 belongs to H. The argument proceeds by showing that any automorphism of P(H) must preserve the collection of singletons (via a combinatorial characterization of additively indecomposable elements) and therefore induces a monoid automorphism of H; since every numerical semigroup admits only the identity automorphism, the induced map on singletons is the identity and the original automorphism is trivial. The same reduction applies to P0(H).

Significance. The result gives a complete determination of the automorphism groups of these power semigroups, linking the additive structure of numerical semigroups to the symmetries of their sumset semigroups. The manuscript supplies a self-contained combinatorial-semigroup argument that reduces the claim to the standard fact that Aut(H) is trivial for numerical semigroups; this reduction is a clear strength of the work.

minor comments (2)
  1. [Introduction] The introduction would benefit from a short explicit example (e.g., H = <2,3>) showing the singletons inside P(H) and confirming that the only automorphism is the identity.
  2. [Section 3] In the section characterizing additively indecomposable elements, the precise combinatorial criterion used to isolate singletons could be stated as a numbered lemma for easier reference in the subsequent reduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes that any automorphism of P(H) preserves singletons via a combinatorial characterization of additively indecomposable elements under sumset, thereby inducing a monoid automorphism of the underlying numerical semigroup H. Since it is a standard, independently verifiable fact that every numerical semigroup admits only the identity automorphism (any such map fixes the minimal generator and preserves the finite gap set), the induced map is forced to be the identity. This reduction uses external combinatorial and semigroup-theoretic facts rather than self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations. The case for P0(H) follows identically when 0 is present. No step in the provided derivation chain reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard definition of numerical semigroups and the induced sumset operation; no free parameters, invented entities, or non-standard axioms are introduced in the abstract.

axioms (1)
  • domain assumption A numerical semigroup is a cofinite subset of the non-negative integers closed under addition.
    This is the foundational definition used to construct P(H) and P0(H).

pith-pipeline@v0.9.0 · 5432 in / 1120 out tokens · 40952 ms · 2026-05-07T10:25:26.056063+00:00 · methodology

discussion (0)

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Reference graph

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