{"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. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.04310","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"}