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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"},"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":"1509.05698","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-18T16:50:35Z","cross_cats_sorted":[],"title_canon_sha256":"09ca7ec99243b5315cf791391706f481eb45a52640b561099536c7d82e552057","abstract_canon_sha256":"1c76c0680c6cf5397301d37aefc9211185c620c83abd25a124d4c2f60580db13"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.579147Z","signature_b64":"T5UPyP0kzwJvZ5/74+dZk03kjeGx9/Ir7Tz9UK1x5FrR7dBJgttWtRzptTzJgOAHQGRhgIOjAY6JBgltRPXZCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32b6e389bb5b2da2bbe248bf8e65a53adcc41d2618bede552117c51b3cb39a3e","last_reissued_at":"2026-05-18T01:32:42.578659Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.578659Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $x(yy)=(xy)y$. 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