# P3100: Embeddability into a finite-group power semigroup

- ID: `P3100`
- Reference: `finite-group-power-semigroup-embeddability`
- Page: https://theoremdb.org/statements/P3100
- Record maturity: Reviewed problem with recorded work

## 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?

### Context

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.

### Problem setup

- **Definition (power semigroup).** For a group G, nonempty subsets multiply by AB={ab:a in A,b in B}.
- **Definition (Classification scope).** Two objects are identified up to semigroup isomorphism unless the statement explicitly asks for an algorithm or equational characterization.
- **Remark.** This is Problem 1.3 in the 2026 Lyapin Notebook; the public formulation fixes the input and equivalence conventions needed for independent review.

### What counts as a solution

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

## 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. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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.
- The release review checked 2 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): Every semigroup is a divisor of a power semigroup over a periodic group, and the divisor version over finite groups is decidable. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): 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](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `finite-group-power-semigroup-embeddability`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://doi.org/10.48550/arXiv.2604.04763
   - 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
   - preprint; primary source; arXiv:2604.04763, checked 2026-08-01; checked 2026-08-01
   - Open copy: https://arxiv.org/abs/2604.04763
   - Source use: original_summary
   - States the numbered open problem, identifies its proposer, and summarizes known special cases.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.1007/s00233-025-10594-3
   - journal_article; primary source; checked 2026-08-01
   - Source 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.