{"paper":{"title":"Isotopisms of quadratic quasigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Jack Allsop","submitted_at":"2025-06-03T05:08:28Z","abstract_excerpt":"A quasigroup is a pair $(Q, \\cdot)$ where $Q$ is a non-empty set and $\\cdot$ is a binary operation on $Q$ such that for every $(u, v) \\in Q^2$ there exists a unique $(x, y) \\in Q^2$ such that $u \\cdot x = v = y \\cdot u$. Let $q$ be an odd prime power, let $\\mathbb{F}_q$ denote the finite field of order $q$, and let $\\mathcal{R}_q$ denote the set of non-zero squares in $\\mathbb{F}_q$. Let $\\{a, b\\} \\subseteq \\mathbb{F}_q$ be such that $\\{ab, (a-1)(b-1)\\} \\subseteq \\mathcal{R}_q$. Let $\\mathcal{Q}_{a, b}$ denote the quadratic quasigroup $(\\mathbb{F}_q, *_{a, b})$ where $*_{a, b}$ is defined by\n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.02446","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.02446/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}