{"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. Note that the first law defines the variety of 3-metabelian group"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.2691","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"}