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From this, we establish Gr\\\"{o}bner-Shirshov bases theory for Lie $\\Omega$-algebras. As applications, we give Gr\\\"{o}bner-Shirshov bases for free $\\lambda$-Rota-Baxter Lie algebras, free modified $\\lambda$-Rota-Baxter Lie algebras and free Nijenhuis Lie algebras and then linear bases of such three free algebras are obtained."},"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":"1604.06675","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-03-24T01:22:08Z","cross_cats_sorted":[],"title_canon_sha256":"e028a3e2b9454dc299ed6685d8382ddec3fa635b59b9d224ce37e167de227fe9","abstract_canon_sha256":"8ba0b60a6d566160398d2ed65de455255ded55e0e7303a331fb441be44e6226a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:28.446342Z","signature_b64":"K/HDmkGB+8kVpRncExAybhdKLjDyv8GewsAvTo4qbsuO/PAIWBZCWCY/kk4Z0KduhC7+8omtVzbArkiVLX4HDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e86145b5b926082bc89d582fb90204dc7a09a4fd9b0cda64050ae9ae8e8cb4f","last_reissued_at":"2026-05-18T01:16:28.445525Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:28.445525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gr\\\"obner-Shirshov bases for Lie $\\Omega$-algebras and free Rota-Baxter Lie algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Jianjun Qiu, Yuqun Chen","submitted_at":"2016-03-24T01:22:08Z","abstract_excerpt":"In this paper, we generalize the Lyndon-Shirshov words to Lyndon-Shirshov $\\Omega$-words on a set $X$ and prove that the set of all non-associative Lyndon-Shirshov $\\Omega$-words forms a linear basis of the free Lie $\\Omega$-algebra on the set $X$. From this, we establish Gr\\\"{o}bner-Shirshov bases theory for Lie $\\Omega$-algebras. As applications, we give Gr\\\"{o}bner-Shirshov bases for free $\\lambda$-Rota-Baxter Lie algebras, free modified $\\lambda$-Rota-Baxter Lie algebras and free Nijenhuis Lie algebras and then linear bases of such three free algebras are obtained."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06675","kind":"arxiv","version":1},"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":"1604.06675","created_at":"2026-05-18T01:16:28.445659+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.06675v1","created_at":"2026-05-18T01:16:28.445659+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.06675","created_at":"2026-05-18T01:16:28.445659+00:00"},{"alias_kind":"pith_short_12","alias_value":"L2DBIW23SJQI","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"L2DBIW23SJQIFPEJ","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"L2DBIW23","created_at":"2026-05-18T12:30:29.479603+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/L2DBIW23SJQIFPEJ2WBPXEBAJX","json":"https://pith.science/pith/L2DBIW23SJQIFPEJ2WBPXEBAJX.json","graph_json":"https://pith.science/api/pith-number/L2DBIW23SJQIFPEJ2WBPXEBAJX/graph.json","events_json":"https://pith.science/api/pith-number/L2DBIW23SJQIFPEJ2WBPXEBAJX/events.json","paper":"https://pith.science/paper/L2DBIW23"},"agent_actions":{"view_html":"https://pith.science/pith/L2DBIW23SJQIFPEJ2WBPXEBAJX","download_json":"https://pith.science/pith/L2DBIW23SJQIFPEJ2WBPXEBAJX.json","view_paper":"https://pith.science/paper/L2DBIW23","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.06675&json=true","fetch_graph":"https://pith.science/api/pith-number/L2DBIW23SJQIFPEJ2WBPXEBAJX/graph.json","fetch_events":"https://pith.science/api/pith-number/L2DBIW23SJQIFPEJ2WBPXEBAJX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L2DBIW23SJQIFPEJ2WBPXEBAJX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L2DBIW23SJQIFPEJ2WBPXEBAJX/action/storage_attestation","attest_author":"https://pith.science/pith/L2DBIW23SJQIFPEJ2WBPXEBAJX/action/author_attestation","sign_citation":"https://pith.science/pith/L2DBIW23SJQIFPEJ2WBPXEBAJX/action/citation_signature","submit_replication":"https://pith.science/pith/L2DBIW23SJQIFPEJ2WBPXEBAJX/action/replication_record"}},"created_at":"2026-05-18T01:16:28.445659+00:00","updated_at":"2026-05-18T01:16:28.445659+00:00"}