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Given a group G we define a double magma (G,*,#) with the commutator operations x * y = [x,y] (= x^-1y^-1xy) and x # y = [y,x]. We show that (G,*,#) is a double magma if and only if G satisfies the commutator laws [x,y;x,z]=1 and [w,x;y,z]^2 = 1. Note that the first law defines the variety of 3-metabelian group"},"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":"1308.2691","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-08-12T20:21:39Z","cross_cats_sorted":[],"title_canon_sha256":"fab381118a5dcd2f580ec1495ffd8591d5c700882afcf553717cf42f1cdfb3f7","abstract_canon_sha256":"020e4217db41391fb273eef9e942e0b0a840a27072052e4d63dc81495229de7f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:16:05.465574Z","signature_b64":"187n3CxAKGtE+9a3QJGxQlikj6lPAfpNQFYm9Dk9bek7hehLSNdsSRdO6jee3ProMRn6IrBTPZWNmgJuLb5vAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"855b420aff90ff2d5995b6dbf1863472b932e2d1c2a2073faefebfc957987ed3","last_reissued_at":"2026-05-18T03:16:05.464879Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:16:05.464879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Constructing Double Magma with Commutation Operations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Charles C. Edmunds","submitted_at":"2013-08-12T20:21:39Z","abstract_excerpt":"A double magma is a nonempty set with two binary operations satisfying the interchange law. We call a double magma proper if the two operations are distinct and commutative if the operations are commutative. A double semigroup is a double magma for which both operations are associative. Given a group G we define a double magma (G,*,#) with the commutator operations x * y = [x,y] (= x^-1y^-1xy) and x # y = [y,x]. We show that (G,*,#) is a double magma if and only if G satisfies the commutator laws [x,y;x,z]=1 and [w,x;y,z]^2 = 1. 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