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As such, $\\beta S$ has a smallest two sided ideal, $K(\\beta S)$. An ultrafilter $p$ on $S$ is \\emph{minimal} if and only if $p \\in K(\\beta S)$.\n  We show that any minimal ultrafilter $p$ factors into a thick part and a syndetic part. That is, there exist filters $\\mathcal F$ and $\\mathcal G$ such that $\\mathcal F$ consists only of thick sets, $\\mathcal G$ consists on"},"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":"1805.07000","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-05-18T00:28:14Z","cross_cats_sorted":["math.GN","math.RA"],"title_canon_sha256":"24c1ec601b50e857e9cbbbcb36d43b8065f1dbe36b3eca9dcc0fd606f82d20ac","abstract_canon_sha256":"3770548acc0bbc48c4f32ce9e752e7d95a1d91013b217f3dc8ef482672c90ae5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:41.404586Z","signature_b64":"T5hBXQifUz0SLrJZPJnI41DhI1dVZEodBTSlkyzyxEBQuWuqseqDakfVPkaqDisNCd/dTMG6A8MiiDdvZYoaBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b109cef88174891e2c9a8c3d4fb49a8ef02707cae27e1bf497152ca19d7c7fb1","last_reissued_at":"2026-05-18T00:15:41.403879Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:41.403879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factoring a minimal ultrafilter into a thick part and a syndetic part","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN","math.RA"],"primary_cat":"math.LO","authors_text":"Neil Hindman, Will Brian","submitted_at":"2018-05-18T00:28:14Z","abstract_excerpt":"Let $S$ be an infinite discrete semigroup. The operation on $S$ extends uniquely to the Stone-\\v{C}ech compactification $\\beta S$ making $\\beta S$ a compact right topological semigroup with $S$ contained in its topological center. As such, $\\beta S$ has a smallest two sided ideal, $K(\\beta S)$. An ultrafilter $p$ on $S$ is \\emph{minimal} if and only if $p \\in K(\\beta S)$.\n  We show that any minimal ultrafilter $p$ factors into a thick part and a syndetic part. 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