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Generalizing this, $< G, \\star >$ is $k$-antiassociative iff for all $x_1, x_2, ... x_k \\in G$, any two distinct expressions made by putting parentheses in $x_1 \\star x_2 \\star x_3 \\star ...x_k$ are never equal.\n  We prove that for every $k \\geq 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equ"},"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":"1410.7501","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2014-10-28T02:55:51Z","cross_cats_sorted":[],"title_canon_sha256":"0d8079ba41ec766ef6e1536dedffa5ebc026ed7b72e6d20e9be6551a26ac733a","abstract_canon_sha256":"a1ad10e7c8c986067d528222897db608465567fb7df9ad70ebf735f3cb40f5c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:17.003234Z","signature_b64":"jsAWB0EQCqe+rENLZLxxaJYPouddmf1yv9aqkJEafU46b2biqa5hKAuc0AiZNEprlGHo28yQtI6uO7MkV5iaBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab2490ed284574790913d45e19eec0d72df48e85c2ea76c18012038f7b600a94","last_reissued_at":"2026-05-18T02:39:17.002821Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:17.002821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Antiassociative Groupoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"David Hobby, Donald Silberger, Milton Braitt","submitted_at":"2014-10-28T02:55:51Z","abstract_excerpt":"Given a groupoid $< G, \\star >$, and $k \\geq 3$, we say that $G$ is antiassociative iff for all $x_1, x_2, x_3 \\in G$, $(x_1 \\star x_2) \\star x_3$ and $x_1 \\star (x_2 \\star x_3)$ are never equal. Generalizing this, $< G, \\star >$ is $k$-antiassociative iff for all $x_1, x_2, ... x_k \\in G$, any two distinct expressions made by putting parentheses in $x_1 \\star x_2 \\star x_3 \\star ...x_k$ are never equal.\n  We prove that for every $k \\geq 3$, there exist finite groupoids that are $k$-antiassociative. 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