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arxiv: 2606.01917 · v1 · pith:U3D5LDX2new · submitted 2026-06-01 · 🧮 math.GR · math.CO· math.RA

Power Semigroups and Two Rigidity Theorems for Groups

Shuolin Liu , Salvatore Tringali This is my paper

Pith reviewed 2026-06-28 12:20 UTC · model grok-4.3

classification 🧮 math.GR math.COmath.RA
keywords power semigroupsrigidity theoremsgroup isomorphismssemigroup isomorphismsfinitary power semigroupsadditive subgroups of rationalsdiophantine approximation
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The pith

If the power semigroup of a group is isomorphic to the power semigroup of a semigroup, then the group and the semigroup are isomorphic.

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

The paper establishes two rigidity results for groups using their power semigroups. It shows that an isomorphism between the full power semigroup of a group and that of any semigroup forces the group and semigroup to be isomorphic. A similar but harder result holds when restricting to finite subsets, but only when the group is an additive subgroup of the rational numbers. These results mean that the group structure is completely recoverable from the semigroup of its subsets under the setwise product.

Core claim

If H is a group and K is a semigroup such that the power semigroup P(H) is isomorphic to P(K), then H is isomorphic to K. The corresponding statement for the finitary power semigroups P_fin(H) and P_fin(K) holds when H is an additive subgroup of the rationals.

What carries the argument

The power semigroup P(H), which is the set of all non-empty subsets of H equipped with the operation A * B = {ab | a in A, b in B}.

Load-bearing premise

The finitary result depends on a special case of a theorem from diophantine approximation about solutions to linear equations over the rationals.

What would settle it

Finding a group H and a semigroup K that are not isomorphic, yet their collections of non-empty subsets with setwise multiplication are isomorphic as semigroups.

read the original abstract

Let $\mathcal P(H)$ be the semigroup obtained by endowing the family of all non-empty subsets of a semigroup $H$ with the setwise operation naturally induced by $H$ on its power set, and denote by $\mathcal P_\text{fin}(H)$ the subsemigroup of $\mathcal P(H)$ consisting of all non-empty finite subsets of $H$. We obtain (as a corollary of a theorem of independent interest) that if $H$ is a group and $K$ is a semigroup, then $\mathcal P(H) \cong \mathcal P(K)$ implies $H \cong K$. The finitary analogue of this statement is considerably more difficult, and we prove it only for $H$ an additive subgroup of the rationals. Most notably, the proof of the second result relies, in a rather circuitous way, on a special case of the Evertse--Schlickewei--Schmidt theorem.

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

2 major / 1 minor

Summary. The manuscript claims that if H is a group and K a semigroup then an isomorphism P(H) ≅ P(K) implies H ≅ K, obtained as a corollary of a theorem of independent interest. It further claims that the finitary analogue holds when H and K are additive subgroups of the rationals, with the proof relying in a circuitous manner on a special case of the Evertse--Schlickewei--Schmidt theorem from diophantine approximation.

Significance. If correct, the results would establish strong rigidity properties showing that the power-semigroup structure determines the underlying group up to isomorphism, with the non-finitary case extending to arbitrary semigroups. The finitary result, though restricted to subgroups of Q, forges a link between semigroup isomorphisms and Diophantine equations that may interest researchers working at the interface of algebra and number theory. No machine-checked proofs or reproducible code are mentioned.

major comments (2)
  1. [Proof of the finitary rigidity theorem] Proof of the finitary rigidity theorem (abstract and the section containing the second result): the reduction that translates an isomorphism of finite power semigroups into an instance of the Evertse--Schlickewei--Schmidt theorem is described as circuitous; the manuscript must supply an explicit account of how every finite-subset equation arising from the semigroup isomorphism is converted into an S-unit equation, and must verify that the argument covers all additive subgroups of Q, including those that are not finitely generated.
  2. [Statement of the main theorem] Statement of the main (non-finitary) rigidity theorem: because the primary claim is presented only as a corollary, the independent theorem on which it rests must be stated in full (including any hypotheses) so that the reader can confirm that the implication P(H) ≅ P(K) o H ≅ K follows without additional restrictions on K.
minor comments (1)
  1. The abstract should briefly name or characterize the independent theorem used for the non-finitary result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address the two major comments below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Proof of the finitary rigidity theorem] Proof of the finitary rigidity theorem (abstract and the section containing the second result): the reduction that translates an isomorphism of finite power semigroups into an instance of the Evertse--Schlickewei--Schmidt theorem is described as circuitous; the manuscript must supply an explicit account of how every finite-subset equation arising from the semigroup isomorphism is converted into an S-unit equation, and must verify that the argument covers all additive subgroups of Q, including those that are not finitely generated.

    Authors: We agree that the reduction step would benefit from greater explicitness. In the revised manuscript we will insert a dedicated subsection that enumerates, for each type of equation arising from an assumed isomorphism of finite power semigroups, the precise sequence of algebraic manipulations that produces the corresponding S-unit equation. We will also add a short paragraph confirming that the argument extends to arbitrary (possibly non-finitely generated) additive subgroups of Q: any such subgroup is the directed union of its finitely generated subgroups, the Evertse--Schlickewei--Schmidt theorem applies uniformly to each finite-rank piece, and the resulting bounds are independent of the rank, so the global conclusion carries over without additional hypotheses. revision: yes

  2. Referee: [Statement of the main theorem] Statement of the main (non-finitary) rigidity theorem: because the primary claim is presented only as a corollary, the independent theorem on which it rests must be stated in full (including any hypotheses) so that the reader can confirm that the implication P(H) ≅ P(K) → H ≅ K follows without additional restrictions on K.

    Authors: We accept the referee’s observation. The revised version will first state the independent theorem in full, with all hypotheses made explicit, and only then derive the corollary that an isomorphism P(H) ≅ P(K) implies H ≅ K whenever H is a group and K is an arbitrary semigroup. This presentation will make the absence of further restrictions on K immediately visible to the reader. revision: yes

Circularity Check

0 steps flagged

No circularity: main result is corollary of independent theorem; finitary case uses external ESS theorem.

full rationale

The abstract explicitly frames the primary rigidity theorem as a corollary of a separate theorem of independent interest, and the finitary case for subgroups of Q is proved via a special case of the external Evertse--Schlickewei--Schmidt theorem from diophantine approximation. No self-citations, self-definitional steps, fitted parameters renamed as predictions, or ansatzes smuggled via prior work by the same authors appear in the provided text. The derivation chain does not reduce any claimed implication to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of the setwise product making P(H) a semigroup, the group axioms for H, and an external diophantine-approximation theorem for the finitary case.

axioms (2)
  • standard math The setwise product of subsets is associative and yields a semigroup.
    Used to define P(H) and P_fin(H).
  • standard math Groups are associative with identity and inverses.
    Invoked when H is stated to be a group.

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Reference graph

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