{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:JWFV3BC5BJHNOJ7AYSYJNGRC3E","short_pith_number":"pith:JWFV3BC5","canonical_record":{"source":{"id":"1407.1967","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-07-08T06:30:05Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7e33197c55d87fc06f0da9a8ee54f5629c76d0e14372a7c57852ec4cd1e8bc17","abstract_canon_sha256":"34f7e860e0665c929197b78678e0d2e7aa1d573386e479eed89b7b319d483030"},"schema_version":"1.0"},"canonical_sha256":"4d8b5d845d0a4ed727e0c4b0969a22d9124af870721a3f3149459c0e9b98bf93","source":{"kind":"arxiv","id":"1407.1967","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.1967","created_at":"2026-05-18T02:03:52Z"},{"alias_kind":"arxiv_version","alias_value":"1407.1967v3","created_at":"2026-05-18T02:03:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.1967","created_at":"2026-05-18T02:03:52Z"},{"alias_kind":"pith_short_12","alias_value":"JWFV3BC5BJHN","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"JWFV3BC5BJHNOJ7A","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"JWFV3BC5","created_at":"2026-05-18T12:28:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:JWFV3BC5BJHNOJ7AYSYJNGRC3E","target":"record","payload":{"canonical_record":{"source":{"id":"1407.1967","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-07-08T06:30:05Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7e33197c55d87fc06f0da9a8ee54f5629c76d0e14372a7c57852ec4cd1e8bc17","abstract_canon_sha256":"34f7e860e0665c929197b78678e0d2e7aa1d573386e479eed89b7b319d483030"},"schema_version":"1.0"},"canonical_sha256":"4d8b5d845d0a4ed727e0c4b0969a22d9124af870721a3f3149459c0e9b98bf93","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:52.799192Z","signature_b64":"MfEdNAACXqsAOg8UpeP/ntGZ+x2JbozY7M1I38rdg2LlfkUWpJGY3snDL8QIsdKSAIlSA4hOJ+8z/l1rGgLDAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d8b5d845d0a4ed727e0c4b0969a22d9124af870721a3f3149459c0e9b98bf93","last_reissued_at":"2026-05-18T02:03:52.798373Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:52.798373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.1967","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:03:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gK+f9c9OzKE8zD7L3K3j1owrks57xc+FGtc/UcEW5wxUtJy8VcjYeboKZgG61WtoAOGittTtRQmnekZqbe+GCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T01:13:43.164437Z"},"content_sha256":"c1a62cc5e4846ced877592ffc8291739370dbac2e95f5276885c66c4df754166","schema_version":"1.0","event_id":"sha256:c1a62cc5e4846ced877592ffc8291739370dbac2e95f5276885c66c4df754166"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:JWFV3BC5BJHNOJ7AYSYJNGRC3E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The system of sets of lengths in Krull monoids under set addition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Alfred Geroldinger (IM), Wolfgang Schmid (LAGA)","submitted_at":"2014-07-08T06:30:05Z","abstract_excerpt":"Let $H$ be a Krull monoid with class group $G$ and suppose that each class contains a prime divisor. Then every element $a \\in H$ has a factorization into irreducible elements, and the set $\\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. We consider the system $\\mathcal L (H) = \\{ \\mathsf L (a) \\mid a \\in H \\}$ of all sets of lengths, and we characterize (in terms of the class group $G$) when $\\mathcal L (H)$ is additively closed under set addition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1967","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:03:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JdW6n/Z9zWihCer/mKddD5HhSPBhXOdglD5SkPMNx60dgECzhxtE9hprxmG8g6dhOsH8GEBDLxVswx9aXmf5CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T01:13:43.164779Z"},"content_sha256":"5345bef56d0041052a77722aad5807629eccaa199765cdce60a5ebda24db4bc0","schema_version":"1.0","event_id":"sha256:5345bef56d0041052a77722aad5807629eccaa199765cdce60a5ebda24db4bc0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/bundle.json","state_url":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-26T01:13:43Z","links":{"resolver":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E","bundle":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/bundle.json","state":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:JWFV3BC5BJHNOJ7AYSYJNGRC3E","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"34f7e860e0665c929197b78678e0d2e7aa1d573386e479eed89b7b319d483030","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-07-08T06:30:05Z","title_canon_sha256":"7e33197c55d87fc06f0da9a8ee54f5629c76d0e14372a7c57852ec4cd1e8bc17"},"schema_version":"1.0","source":{"id":"1407.1967","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.1967","created_at":"2026-05-18T02:03:52Z"},{"alias_kind":"arxiv_version","alias_value":"1407.1967v3","created_at":"2026-05-18T02:03:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.1967","created_at":"2026-05-18T02:03:52Z"},{"alias_kind":"pith_short_12","alias_value":"JWFV3BC5BJHN","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"JWFV3BC5BJHNOJ7A","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"JWFV3BC5","created_at":"2026-05-18T12:28:35Z"}],"graph_snapshots":[{"event_id":"sha256:5345bef56d0041052a77722aad5807629eccaa199765cdce60a5ebda24db4bc0","target":"graph","created_at":"2026-05-18T02:03:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $H$ be a Krull monoid with class group $G$ and suppose that each class contains a prime divisor. Then every element $a \\in H$ has a factorization into irreducible elements, and the set $\\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. We consider the system $\\mathcal L (H) = \\{ \\mathsf L (a) \\mid a \\in H \\}$ of all sets of lengths, and we characterize (in terms of the class group $G$) when $\\mathcal L (H)$ is additively closed under set addition.","authors_text":"Alfred Geroldinger (IM), Wolfgang Schmid (LAGA)","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-07-08T06:30:05Z","title":"The system of sets of lengths in Krull monoids under set addition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1967","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c1a62cc5e4846ced877592ffc8291739370dbac2e95f5276885c66c4df754166","target":"record","created_at":"2026-05-18T02:03:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"34f7e860e0665c929197b78678e0d2e7aa1d573386e479eed89b7b319d483030","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-07-08T06:30:05Z","title_canon_sha256":"7e33197c55d87fc06f0da9a8ee54f5629c76d0e14372a7c57852ec4cd1e8bc17"},"schema_version":"1.0","source":{"id":"1407.1967","kind":"arxiv","version":3}},"canonical_sha256":"4d8b5d845d0a4ed727e0c4b0969a22d9124af870721a3f3149459c0e9b98bf93","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d8b5d845d0a4ed727e0c4b0969a22d9124af870721a3f3149459c0e9b98bf93","first_computed_at":"2026-05-18T02:03:52.798373Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:03:52.798373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MfEdNAACXqsAOg8UpeP/ntGZ+x2JbozY7M1I38rdg2LlfkUWpJGY3snDL8QIsdKSAIlSA4hOJ+8z/l1rGgLDAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:03:52.799192Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.1967","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c1a62cc5e4846ced877592ffc8291739370dbac2e95f5276885c66c4df754166","sha256:5345bef56d0041052a77722aad5807629eccaa199765cdce60a5ebda24db4bc0"],"state_sha256":"902ebb9c2131830a3b6e52b9e653185c9d3acb1646baa5911beb5ae03b42a886"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o2d2djMZtUbbdzN+VTHB9+u7qRvk4WQ+n8eyac0Wm2mLIMJ1dKOI/iLgICswKClAj1qqhrpm22wNP87jC9pVBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T01:13:43.166696Z","bundle_sha256":"179a76ba25ebe02d50972639f9b0a553ab99695e2ea91e11756248f572cadd7a"}}