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A set-theoretical solution of the pentagon equation on $M$ is a map $s:M\\times M\\longrightarrow M\\times M$ such that \\begin{equation*} s_{23}\\, s_{13}\\, s_{12}=s_{12}\\, s_{23}, \\end{equation*} where $s_{12}=s\\times id_M$, $s_{23}=id_M \\times s$ and $s_{13}=(id_M \\times \\tau) s_{12}(id_M \\times \\tau)$, and $\\tau$ is the flip map, i.e., the permutation on $M\\times M$ given by $\\tau(x,y)=(y,x)$, for all $x,y\\in M$. In this paper we give a complete description of the set-theoretical solutions of the form $s(x,y)=(x\\cdot y , x\\ast y)$ when either $(M,\\cdot)$ or $(M,\\ast)$ is a gro"},"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":"1902.04310","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2019-02-12T10:06:41Z","cross_cats_sorted":[],"title_canon_sha256":"e4c93784eef1d6e7e0d0fb32b060050a81c95fb6538936ffd4dd7d82173c489e","abstract_canon_sha256":"3d0ca002ece77c9be0f0909449730c06b42725dd7333a8b512683a3dd7e3e23f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:12.609131Z","signature_b64":"t4OyMtoQbKqmZMKguIGg4rqq89IWZFkwAbLGJT6HHR8XZfYNFGfPWozE0omXgMAcdcAmNa4xfwLgDFpJIOl6Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b23f0bc26b0d1626424f1ffdd481ba8f4d6e1aaab6abd63cf48877105a553cc","last_reissued_at":"2026-05-17T23:54:12.608533Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:12.608533Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Set-theoretical solutions of the pentagon equation on groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Francesco Catino, Maria Maddalena Miccoli, Marzia Mazzotta","submitted_at":"2019-02-12T10:06:41Z","abstract_excerpt":"Let $M$ be a set. 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