{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VCBYLECRMK7BZS5DX3TGMBBMFT","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":"610c631ecac938e71f9b2ec50f0342cc875ae705b7bff1c961f9bf2959393ae5","cross_cats_sorted":["math.CO","math.NT","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-12-09T18:16:32Z","title_canon_sha256":"885b0d0893ddad6bf42ab0b2d066c31beb62c62ffb47f7046ade2e249d72323d"},"schema_version":"1.0","source":{"id":"1612.03116","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.03116","created_at":"2026-05-18T00:16:09Z"},{"alias_kind":"arxiv_version","alias_value":"1612.03116v3","created_at":"2026-05-18T00:16:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.03116","created_at":"2026-05-18T00:16:09Z"},{"alias_kind":"pith_short_12","alias_value":"VCBYLECRMK7B","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VCBYLECRMK7BZS5D","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VCBYLECR","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:d8aea64e0d572c86e15b5765994d1bbb3f2653f9e5124498b80410594931ef03","target":"graph","created_at":"2026-05-18T00:16:09Z","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 commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k \\in \\mathbb N$, let $\\mathscr U_k (H)$ denote the set of all $\\ell \\in \\mathbb N$ with the property that there are atoms $u_1, \\ldots, u_k, v_1, \\ldots, v_{\\ell}$ such that $u_1 \\cdot \\ldots \\cdot u_k = v_1 \\cdot \\ldots \\cdot v_{\\ell}$ (thus, $\\mathscr U_k (H)$ is the union of all sets of lengths containing $k$).\n  The Structure Theorem for Unions states that, for all sufficiently large $k$, the sets $\\mathscr U_k (H)$ are almost arithme","authors_text":"Alfred Geroldinger, Florian Kainrath, Salvatore Tringali, Yushuang Fan","cross_cats":["math.CO","math.NT","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-12-09T18:16:32Z","title":"Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.03116","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":"sha256:443584754b5982d60cf987a5933ca77f9ea44244515b9903bc25e72e23a77056","target":"record","created_at":"2026-05-18T00:16:09Z","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":"610c631ecac938e71f9b2ec50f0342cc875ae705b7bff1c961f9bf2959393ae5","cross_cats_sorted":["math.CO","math.NT","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-12-09T18:16:32Z","title_canon_sha256":"885b0d0893ddad6bf42ab0b2d066c31beb62c62ffb47f7046ade2e249d72323d"},"schema_version":"1.0","source":{"id":"1612.03116","kind":"arxiv","version":3}},"canonical_sha256":"a88385905162be1ccba3bee666042c2ce7ccca413e7f894742b678826048659c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a88385905162be1ccba3bee666042c2ce7ccca413e7f894742b678826048659c","first_computed_at":"2026-05-18T00:16:09.446660Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:09.446660Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7IkKK4MzAInPXGfwX7ccRCkDOfvHqVaDDa3vGdIoIsv0OR5N1F3YHBvxtVyEB2fLY8UteRXIYFjTE2LWqtH7AA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:09.447589Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.03116","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:443584754b5982d60cf987a5933ca77f9ea44244515b9903bc25e72e23a77056","sha256:d8aea64e0d572c86e15b5765994d1bbb3f2653f9e5124498b80410594931ef03"],"state_sha256":"639bbe346208399e0a943984d0b4a1784c43003d47d4382aac0692aa1da7615d"}