{"paper":{"title":"Products of all elements in a loop and a framework for non-associative analogues of the Hall-Paige conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Kyle Pula","submitted_at":"2010-08-04T05:17:26Z","abstract_excerpt":"For a finite loop $Q$, let $P (Q)$ be the set of elements that can be represented as a product containing each element of $Q$ precisely once. Motivated by the recent proof of the Hall-Paige conjecture, we prove several universal implications between the following conditions: (A) $Q$ has a complete mapping, i.e. the multiplication table of $Q$ has a transversal, (B) there is no $N \\normal Q$ such that $|N|$ is odd and $Q/N \\cong \\ZZ_{2^m}$ for $m \\geq 1$, and (C) $P(Q)$ intersects the associator subloop of $Q$. We prove $(A) \\implies (C)$ and $(B) \\iff (C)$ and show that when $Q$ is a group, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.0699","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"}