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Denote by $T(X)$ the semigroup of full transformations on a finite set $X$. Let $J$ be any ideal of $T(X)$ such that $J$ is different from the ideal of constant transformations on $X$. We prove that if $|X|\\geq4$, then, with a few exceptions, the diameter of $\\cg(J)$ is 5. On the other hand, we prove that for every positive integer $n$, there exists a semigr"},"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":"1003.2809","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-03-14T20:14:23Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0608a2ac5e97622ac8972795829d4ba65547c454d5031f5994f8b04f73ce9b66","abstract_canon_sha256":"82e8dc9aa3df752188be3e5018b238a329d22391d8d4b48c171b9f963a7c5574"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:17.041967Z","signature_b64":"9XRl/A7x6dsqX9ET3S3QaxDCw+QQ35FWJaq2lb2463WKLSNid4Pn6hdHghDD6qaOeOWmPfZjRzUvAXUmPF/aCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af1b80f33892535fa9c14590ac4d6a3f28523b8dd448047e7fccb7d36fb1afa6","last_reissued_at":"2026-05-18T04:15:17.041275Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:17.041275Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal paths in the commuting graphs of semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Janusz Konieczny, Joao Araujo, Michael Kinyon","submitted_at":"2010-03-14T20:14:23Z","abstract_excerpt":"Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted $\\cg(S)$, is the graph whose vertices are the non-central elements of $S$ and whose edges are the sets $\\{a,b\\}$ of vertices such that $a\\ne b$ and $ab=ba$. Denote by $T(X)$ the semigroup of full transformations on a finite set $X$. Let $J$ be any ideal of $T(X)$ such that $J$ is different from the ideal of constant transformations on $X$. We prove that if $|X|\\geq4$, then, with a few exceptions, the diameter of $\\cg(J)$ is 5. 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