[#P3100] Embeddability into a finite-group power semigroup

Problem. Is there an algorithm that, given the multiplication table of a finite semigroup \(S\), decides whether \(S\) embeds into \(\mathcal P^*(G)\) for some finite group \(G\), where \(\mathcal P^*(G)\) is the semigroup of nonempty subsets of \(G\) under setwise multiplication?

Neutral schematic of a finite semigroup multiplication diagram illustrating power semigroup. — The objects and operations appearing in Embeddability into a finite-group power semigroup.

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1Context

Known frontier: Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable. Open boundary: Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.

2Problem setup

Definition 1 (power semigroup). For a group G, nonempty subsets multiply by AB={ab:a in A,b in B}.

Definition 2 (Classification scope). Two objects are identified up to semigroup isomorphism unless the statement explicitly asks for an algorithm or equational characterization.

Remark 1. This is Problem 1.3 in the 2026 Lyapin Notebook; the public formulation fixes the input and equivalence conventions needed for independent review.

3What counts as a solution

Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable. Exact unresolved remainder: Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.^[1][2]

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable. Exact unresolved remainder: Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable. Exact unresolved remainder: Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Embeddability into a finite-group power semigroup. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable. The exact unresolved remainder is: Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists. A complete resolution must meet the following acceptance conditions: - Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists. Open full record →
Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable.^[1][2] Relevance to this problem Locates the present research frontier immediately below Embeddability into a finite-group power semigroup. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable. This leaves the following boundary unresolved: Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there.^[1][2] Relevance to this problem Documents why Embeddability into a finite-group power semigroup was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across source collection and its cited primary literature, Crossref/arXiv or MathSciNet stable records, TheoremDB published, prospecting, formal-conjecture, packet, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - "Lyapin notebook" "Problem 1.3" - "Embeddability into a finite-group power semigroup" - site:arxiv.org semigroup "power semigroup" open problem The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there. Related: Claim 1 Open full record →
Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists. Relevance to this problem Turns the remaining uncertainty in Embeddability into a finite-group power semigroup into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable. Exact unresolved remainder: Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.

Equivalent-formulation queries: "Lyapin notebook" "Problem 1.3"; "Embeddability into a finite-group power semigroup"; site:arxiv.org semigroup "power semigroup" open problem
Strongest checked neighboring result: Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable.
Exact unresolved remainder: Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.

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ProblemEmbeddability into a finite-group power semigroup

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“Embeddability into a finite-group power semigroup.” TheoremDB. P3100. Problem statement; statement text SHA-256 e43190845ef3beb26aba1fc335d8eec0a3c68c9da87da31eb019b3c03da959e9. https://theoremdb.org/statement/?ref=P3100

BibTeX

@misc{theoremdb-problem-e43190845ef3beb26aba1fc335d8eec0a3c68c9da87da31eb019b3c03da959e9,
  title = {{Embeddability into a finite-group power semigroup}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 e43190845ef3beb26aba1fc335d8eec0a3c68c9da87da31eb019b3c03da959e9},
  url = {https://theoremdb.org/statement/?ref=P3100}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

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1References

Packet source. Bershadsky, S., Kublanovsky, S., and Mashevitzky, G., “The Lyapin's notebook: a collection of unsolved problems in Semigroup Theory”. arXiv (2026). DOI 10.48550/arXiv.2604.04763. Problem 1.3 and its immediately following status paragraph. ↗ open copy ↗preprint · primary source · arXiv:2604.04763, checked 2026-08-01 · checked 2026-08-01Source use: original summary.States the numbered open problem, identifies its proposer, and summarizes known special cases.Also cited at S. Bershadsky, S. Kublanovsky, and G. Mashevitzky, “The Lyapin’s notebook: a collection of unsolved problems in Semigroup Theory,” arXiv:2604.04763 (2026). Problem 1.3 and its immediately following status paragraph.Source used to assess the problem's recorded status.For Embeddability into a finite-group power semigroup: This is the dated publication status for the canonical target Embeddability into a finite-group power semigroup.Source named by the research packet.
S. G. Bershadsky and S. I. Kublanovsky, “On power semigroups of groups”. Semigroup Forum 111(3) (2025), 543-560. DOI 10.1007/s00233-025-10594-3. Main embeddability theorems and open periodic-group boundary. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves the quoted divisor and embedding results and develops the precise power-semigroup setting.Source used to assess the problem's recorded status.For Embeddability into a finite-group power semigroup: Proves the quoted divisor and embedding results and develops the precise power-semigroup setting.

Original TheoremDB statement and summary based on citation-only scholarly sources; no source prose, proof, table, code, or figure is reproduced.

[#P3100] Embeddability into a finite-group power semigroup

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