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We secondly establish Composition-Diamond lemma for $L$-algebras. As applications, we give Gr\\\"{o}bner-Shirshov bases of the free dialgebra and the free product of two $L$-algebras, and then we show four embedding theorems of $L$-algebras: 1) Every countably generated $L$-algebra  can be embedded into a two-generated $L$-algebra. 2) Every $"},"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":"1005.0118","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-02T02:20:47Z","cross_cats_sorted":[],"title_canon_sha256":"ad306063bb20eabda0021627ff8b78de510581282308876019751a2165d5a2c1","abstract_canon_sha256":"10f996d2924aa8d6605475887d916acfb4507e32afd0bdf3612c606da59baca4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:09.469343Z","signature_b64":"XfjqlXdHZJdSiGx5Qz54Tboyedtv6pR071tJ/sjbmBira3q1dqvLrT9BvS4Opdwx+Y75DQKR4EsZJXd/Z6HAAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08826a96e6efb1c7b77c2bcc95e16bb3430b529384d425ccadd281bb237ded8a","last_reissued_at":"2026-05-18T02:24:09.468663Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:09.468663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gr\\\"obner-Shirshov bases for $L$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Jiapeng Huang, L.A. Bokut, Yuqun Chen","submitted_at":"2010-05-02T02:20:47Z","abstract_excerpt":"In this paper, we firstly establish Composition-Diamond lemma for $\\Omega$-algebras. We give a Gr\\\"{o}bner-Shirshov basis of the free $L$-algebra as a quotient algebra of a free $\\Omega$-algebra, and then the normal form of the free $L$-algebra is obtained. We secondly establish Composition-Diamond lemma for $L$-algebras. 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