{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2004:CRWYOUPR2XDUKZCWR6VIEY55I7","short_pith_number":"pith:CRWYOUPR","schema_version":"1.0","canonical_sha256":"146d8751f1d5c74564568faa8263bd47d0206c58bc60f4f2c2e7c56141ac3f03","source":{"kind":"arxiv","id":"math/0411268","version":2},"attestation_state":"computed","paper":{"title":"Associativity in multary quasigroups: The way of biased expansions","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Thomas Zaslavsky","submitted_at":"2004-11-11T20:39:05Z","abstract_excerpt":"A \"biased expansion\" of a graph is a kind of branched covering graph with additional structure related to combinatorial homotopy of circles. Some but not all biased expansions are constructed from groups (\"group expansions\"); these include all biased expansions of complete graphs (assuming order at least four), which correspond to Dowling's lattices of a group and encode an iterated group operation. A biased expansion of a circle with chords encodes a multary (polyadic, n-ary) quasigroup, the chords corresponding to factorizations, i.e., associative structure. We show that any biased expansion"},"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":"math/0411268","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2004-11-11T20:39:05Z","cross_cats_sorted":[],"title_canon_sha256":"f9a2b8eb4d6550e9e9fd09828df07637a5c2536435fda8bf89eea23ed7b4a261","abstract_canon_sha256":"0a07dc9b82506cb0a9afd76f3eaf0f653dd1d7400d73fe18faf200888163dac8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:05.577789Z","signature_b64":"SUP5p/vLK+RhD9/LrIbzwF1w75SLyRe+DZrvzTxYaqEHbtgXXYkqlSEyW3X20eBttynAQb0ZHZ4c/Tyn+dJBAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"146d8751f1d5c74564568faa8263bd47d0206c58bc60f4f2c2e7c56141ac3f03","last_reissued_at":"2026-05-18T01:02:05.577125Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:05.577125Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Associativity in multary quasigroups: The way of biased expansions","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Thomas Zaslavsky","submitted_at":"2004-11-11T20:39:05Z","abstract_excerpt":"A \"biased expansion\" of a graph is a kind of branched covering graph with additional structure related to combinatorial homotopy of circles. Some but not all biased expansions are constructed from groups (\"group expansions\"); these include all biased expansions of complete graphs (assuming order at least four), which correspond to Dowling's lattices of a group and encode an iterated group operation. A biased expansion of a circle with chords encodes a multary (polyadic, n-ary) quasigroup, the chords corresponding to factorizations, i.e., associative structure. We show that any biased expansion"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0411268","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":"math/0411268","created_at":"2026-05-18T01:02:05.577226+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0411268v2","created_at":"2026-05-18T01:02:05.577226+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0411268","created_at":"2026-05-18T01:02:05.577226+00:00"},{"alias_kind":"pith_short_12","alias_value":"CRWYOUPR2XDU","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_16","alias_value":"CRWYOUPR2XDUKZCW","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_8","alias_value":"CRWYOUPR","created_at":"2026-05-18T12:25:52.687210+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/CRWYOUPR2XDUKZCWR6VIEY55I7","json":"https://pith.science/pith/CRWYOUPR2XDUKZCWR6VIEY55I7.json","graph_json":"https://pith.science/api/pith-number/CRWYOUPR2XDUKZCWR6VIEY55I7/graph.json","events_json":"https://pith.science/api/pith-number/CRWYOUPR2XDUKZCWR6VIEY55I7/events.json","paper":"https://pith.science/paper/CRWYOUPR"},"agent_actions":{"view_html":"https://pith.science/pith/CRWYOUPR2XDUKZCWR6VIEY55I7","download_json":"https://pith.science/pith/CRWYOUPR2XDUKZCWR6VIEY55I7.json","view_paper":"https://pith.science/paper/CRWYOUPR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0411268&json=true","fetch_graph":"https://pith.science/api/pith-number/CRWYOUPR2XDUKZCWR6VIEY55I7/graph.json","fetch_events":"https://pith.science/api/pith-number/CRWYOUPR2XDUKZCWR6VIEY55I7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CRWYOUPR2XDUKZCWR6VIEY55I7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CRWYOUPR2XDUKZCWR6VIEY55I7/action/storage_attestation","attest_author":"https://pith.science/pith/CRWYOUPR2XDUKZCWR6VIEY55I7/action/author_attestation","sign_citation":"https://pith.science/pith/CRWYOUPR2XDUKZCWR6VIEY55I7/action/citation_signature","submit_replication":"https://pith.science/pith/CRWYOUPR2XDUKZCWR6VIEY55I7/action/replication_record"}},"created_at":"2026-05-18T01:02:05.577226+00:00","updated_at":"2026-05-18T01:02:05.577226+00:00"}