{"paper":{"title":"Powers and alternative laws","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Nicholas Ormes, Petr Vojt\\v{e}chovsk\\'y","submitted_at":"2015-09-18T16:50:35Z","abstract_excerpt":"A groupoid is alternative if it satisfies the alternative laws $x(xy)=(xx)y$ and $x(yy)=(xy)y$. These laws induce four partial maps on $\\mathbb{N}^+\\times \\mathbb{N}^+$, $(r,\\,s)\\mapsto (2r,\\,s-r)$, $(r-s,\\,2s)$, $(r/2,\\,s+r/2)$, $(r+s/2,\\,s/2)$ that taken together form a dynamical system. We describe the orbits of this dynamical system, which allows us to show that $n$th powers in a free alternative groupoid on one generator are well-defined if and only if $n\\le 5$. We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inver"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05698","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"}