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We also state several conditions that are equivalent to achieving the van Rees bound. One of these is that the Cayley table of a loop achieves the van Rees bound if and only if every loop isotope has exponent $3$. We call such loops \\emph{van Rees loops} and show that they form an equationally defined variety.\n  We also show that (1) In a va"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1103.0054","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-02-28T23:43:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4274ad77e96417ae3c80fe502e79ca7710f537bc642930bc76e5aae56808cee4","abstract_canon_sha256":"87bfc94443045bbecf20197e262765a6c116a72824ddede177a558456dc96558"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:39.340967Z","signature_b64":"zlz/oL6AIvrAk+/g/aHkmrWWipHEtiPyO9r5bPDWo9ZyojI22o8JNaQvX6dvZiZiZPcva26HBMbLqL0TuMaTBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67d6020e88845b36449296c689df9b6b957a5688c50968cab6cda961d68ca8ce","last_reissued_at":"2026-05-18T01:23:39.340348Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:39.340348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Loops with exponent three in all isotopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Ian M. Wanless, Michael Kinyon","submitted_at":"2011-02-28T23:43:57Z","abstract_excerpt":"It was shown by van Rees \\cite{vR} that a latin square of order $n$ has at most $n^2(n-1)/18$ latin subsquares of order $3$. He conjectured that this bound is only achieved if $n$ is a power of $3$. We show that it can only be achieved if $n\\equiv3\\bmod6$. We also state several conditions that are equivalent to achieving the van Rees bound. One of these is that the Cayley table of a loop achieves the van Rees bound if and only if every loop isotope has exponent $3$. We call such loops \\emph{van Rees loops} and show that they form an equationally defined variety.\n  We also show that (1) In a va"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0054","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1103.0054","created_at":"2026-05-18T01:23:39.340461+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.0054v2","created_at":"2026-05-18T01:23:39.340461+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.0054","created_at":"2026-05-18T01:23:39.340461+00:00"},{"alias_kind":"pith_short_12","alias_value":"M7LAEDUIQRNT","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_16","alias_value":"M7LAEDUIQRNTMRES","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_8","alias_value":"M7LAEDUI","created_at":"2026-05-18T12:26:34.985390+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/M7LAEDUIQRNTMRESS3DITX43NO","json":"https://pith.science/pith/M7LAEDUIQRNTMRESS3DITX43NO.json","graph_json":"https://pith.science/api/pith-number/M7LAEDUIQRNTMRESS3DITX43NO/graph.json","events_json":"https://pith.science/api/pith-number/M7LAEDUIQRNTMRESS3DITX43NO/events.json","paper":"https://pith.science/paper/M7LAEDUI"},"agent_actions":{"view_html":"https://pith.science/pith/M7LAEDUIQRNTMRESS3DITX43NO","download_json":"https://pith.science/pith/M7LAEDUIQRNTMRESS3DITX43NO.json","view_paper":"https://pith.science/paper/M7LAEDUI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.0054&json=true","fetch_graph":"https://pith.science/api/pith-number/M7LAEDUIQRNTMRESS3DITX43NO/graph.json","fetch_events":"https://pith.science/api/pith-number/M7LAEDUIQRNTMRESS3DITX43NO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/M7LAEDUIQRNTMRESS3DITX43NO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/M7LAEDUIQRNTMRESS3DITX43NO/action/storage_attestation","attest_author":"https://pith.science/pith/M7LAEDUIQRNTMRESS3DITX43NO/action/author_attestation","sign_citation":"https://pith.science/pith/M7LAEDUIQRNTMRESS3DITX43NO/action/citation_signature","submit_replication":"https://pith.science/pith/M7LAEDUIQRNTMRESS3DITX43NO/action/replication_record"}},"created_at":"2026-05-18T01:23:39.340461+00:00","updated_at":"2026-05-18T01:23:39.340461+00:00"}