{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1539",
  "content_hash": null,
  "slug": "finite-group-power-semigroup-embeddability-claim-status-20260801",
  "type": "claim",
  "title": "Current status and exact unresolved remainder",
  "summary": "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.",
  "relevance": "This is the dated publication status for the canonical target Embeddability into a finite-group power semigroup.",
  "relevance_source": "recorded",
  "body": "The problem was checked as open on 2026-08-01.\n\nThe 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.\n\nThe exact unresolved remainder is: Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.\n\nA complete resolution must meet the following acceptance conditions:\n- Give a terminating correct algorithm for direct embeddability, or prove that no such algorithm exists.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.48550/arXiv.2604.04763",
      "locator": "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"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.48550/arXiv.2604.04763",
    "locator": "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"
  },
  "relations": [
    {
      "slug": "R1538",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1536",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1537",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "finite-group-power-semigroup-embeddability",
      "title": "finite group power semigroup embeddability",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}