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We say that $G$ synchronizes $f$ if the semigroup $\\langle G,f\\rangle$ contains a constant map.\n  The first author has conjectured that a primitive group synchronizes any map whose kernel is non-uniform. Rystsov proved one instance of this conjecture, namely, degree $n$ primitive groups synchronize maps of rank $n-1$ (thus, maps with kernel type $(2,1,\\ldots,1)$). We prove some extensions of Rystsov's result, including this: a primitive group synchronizes eve"},"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":"1306.4827","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-20T11:38:19Z","cross_cats_sorted":[],"title_canon_sha256":"4fba6c4dde9d29c325126ec907ababd23aa1936860e6e00122c7fc58ddcffe58","abstract_canon_sha256":"df522f39e6b14fcbb2f818080460ffebee817389cd56425600f66bb0598e99bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:01:13.679261Z","signature_b64":"JgCy9hQPAy+HIdovXNUtbXlGuIwtppfrcAIhVIGEjE3vJ46dQvxEg/a5LN3tfMB1xhJY6Us/KOk54erNORpQCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2a7ce4431f4fba6ea140d313a35d287d0ab12f08588ec48d31ce266e1d7d553","last_reissued_at":"2026-05-18T03:01:13.678715Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:01:13.678715Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Primitive Groups Synchronize Non-uniform Maps of Extreme Ranks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jo\\~ao Ara\\'ujo, Peter J. Cameron","submitted_at":"2013-06-20T11:38:19Z","abstract_excerpt":"Let $\\Omega$ be a set of cardinality $n$, $G$ a permutation group on $\\Omega$, and $f:\\Omega\\to\\Omega$ a map which is not a permutation. We say that $G$ synchronizes $f$ if the semigroup $\\langle G,f\\rangle$ contains a constant map.\n  The first author has conjectured that a primitive group synchronizes any map whose kernel is non-uniform. Rystsov proved one instance of this conjecture, namely, degree $n$ primitive groups synchronize maps of rank $n-1$ (thus, maps with kernel type $(2,1,\\ldots,1)$). 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