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Conversely, if $\\alpha$ is a congruence on a commutative semigroup $C$ such that the factor semigroup $S=C/\\alpha$ satisfies Condition $(*)$ then there is an ide"},"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":"1508.07430","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-08-29T10:22:57Z","cross_cats_sorted":["math.NT","math.RA"],"title_canon_sha256":"c91d480c101b4958fa6e4ca597dbdc3d48dca5c9aafa3c8a146b3441fbeffbb8","abstract_canon_sha256":"b20754ada4f841d7d2b75d83cdd0a380691385f4b2cfec283d8fc9e92ef3fedc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:32.321162Z","signature_b64":"tnweLzXsxS3sFU/87A8BO0p3k929e/3VPCLFBol26YjSuHc1FnzPBrSDGxIYYk3bgu6mvDmlzi9xf08zDCdMDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cba47e6266aeca6c7a88cceb5d9e4a8b142375fa37ffb1bac2191aff53f99a30","last_reissued_at":"2026-05-18T01:34:32.320393Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:32.320393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Separators of Ideals in Multiplicative Semigroups of Unique Factorization Domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.RA"],"primary_cat":"math.GR","authors_text":"Attila Nagy","submitted_at":"2015-08-29T10:22:57Z","abstract_excerpt":"In this paper we show that if $I$ is an ideal of a commutative semigroup $C$ such that the separator $SepI$ of $I$ is not empty then the factor semigroup $S=C/P_I$ ($P_I$ is the principal congruence on $C$ defined by $I$) satisfies Condition $(*)$: $S$ is a commutative monoid with a zero; The annihilator $A(s)$ of every non identity element $s$ of $S$ contains a non zero element of $S$; $A(s)=A(t)$ implies $s=t$ for every $s, t\\in S$. 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