{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:3KMWANBXEFLSDNFCU6PZDQTMDE","short_pith_number":"pith:3KMWANBX","canonical_record":{"source":{"id":"1512.01513","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-12-04T19:11:12Z","cross_cats_sorted":[],"title_canon_sha256":"a405efa4340ed638ddc3db160b8e28b330b8b50e23ef7603990beb25b83f3b63","abstract_canon_sha256":"682e2438ed15bbdf0b2ff63dc946e293ffb3a9d6f74e035dd6a05d098e4d02da"},"schema_version":"1.0"},"canonical_sha256":"da99603437215721b4a2a79f91c26c191f7db54bc2f914ea78e00dd403b7d27c","source":{"kind":"arxiv","id":"1512.01513","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.01513","created_at":"2026-05-18T01:11:17Z"},{"alias_kind":"arxiv_version","alias_value":"1512.01513v2","created_at":"2026-05-18T01:11:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01513","created_at":"2026-05-18T01:11:17Z"},{"alias_kind":"pith_short_12","alias_value":"3KMWANBXEFLS","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"3KMWANBXEFLSDNFC","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"3KMWANBX","created_at":"2026-05-18T12:29:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:3KMWANBXEFLSDNFCU6PZDQTMDE","target":"record","payload":{"canonical_record":{"source":{"id":"1512.01513","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-12-04T19:11:12Z","cross_cats_sorted":[],"title_canon_sha256":"a405efa4340ed638ddc3db160b8e28b330b8b50e23ef7603990beb25b83f3b63","abstract_canon_sha256":"682e2438ed15bbdf0b2ff63dc946e293ffb3a9d6f74e035dd6a05d098e4d02da"},"schema_version":"1.0"},"canonical_sha256":"da99603437215721b4a2a79f91c26c191f7db54bc2f914ea78e00dd403b7d27c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:17.886100Z","signature_b64":"u6Sx9C365Oiikg0ViWQ0lMV/IeofZ4wTD3/nVaP00kJJLkeanLyJsWuTu2JSkxhGys1hMoxu0IjMyNdSyKTQCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da99603437215721b4a2a79f91c26c191f7db54bc2f914ea78e00dd403b7d27c","last_reissued_at":"2026-05-18T01:11:17.885428Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:17.885428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.01513","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:11:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L4NwQzS9f44NuXUUk8qCWbxMcWPPGLcQ8VnGLvVGbIHEa/hoam99CdtMfoOsq/OnwYfN/BI+BbuNx4XOFT1eBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T01:44:05.068234Z"},"content_sha256":"a4290ed292e306f4ab45517a2d3f819b919dbac37cbb0e1dd1d1e0888d3c6b98","schema_version":"1.0","event_id":"sha256:a4290ed292e306f4ab45517a2d3f819b919dbac37cbb0e1dd1d1e0888d3c6b98"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:3KMWANBXEFLSDNFCU6PZDQTMDE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proportionally modular affine semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"A. Vigneron-Tenorio, J. I. Garc\\'ia-Garc\\'ia, M. A. Moreno-Fr\\'ias","submitted_at":"2015-12-04T19:11:12Z","abstract_excerpt":"This work introduces a new kind of semigroup of $\\N^p$ called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We prove they are finitely generated and we give an algorithm to compute their minimal generating sets. We also specialise on the case $p=2$. For this case, we provide a faster algorithm to compute their minimal system of generators and we prove they are Cohen-Macaulay and Buchsbaum. Besides, the Gorenstein property is charactized, and their (minimal) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01513","kind":"arxiv","version":2},"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":"2026-05-18T01:11:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UbBKbSJTerX36rrakKUvYQJnVRV+u1Wm19x0qVrxYUXZOSh+TeacfuNz8RjpsKsr/zWWE7i0XF6R8XEnOdvyDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T01:44:05.068609Z"},"content_sha256":"f490fa44dbb328b6b176e6ba450c8b3a3dd46af0d3d5fe79302b77c4f9f9d70d","schema_version":"1.0","event_id":"sha256:f490fa44dbb328b6b176e6ba450c8b3a3dd46af0d3d5fe79302b77c4f9f9d70d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3KMWANBXEFLSDNFCU6PZDQTMDE/bundle.json","state_url":"https://pith.science/pith/3KMWANBXEFLSDNFCU6PZDQTMDE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3KMWANBXEFLSDNFCU6PZDQTMDE/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":"2026-06-05T01:44:05Z","links":{"resolver":"https://pith.science/pith/3KMWANBXEFLSDNFCU6PZDQTMDE","bundle":"https://pith.science/pith/3KMWANBXEFLSDNFCU6PZDQTMDE/bundle.json","state":"https://pith.science/pith/3KMWANBXEFLSDNFCU6PZDQTMDE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3KMWANBXEFLSDNFCU6PZDQTMDE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:3KMWANBXEFLSDNFCU6PZDQTMDE","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":"682e2438ed15bbdf0b2ff63dc946e293ffb3a9d6f74e035dd6a05d098e4d02da","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-12-04T19:11:12Z","title_canon_sha256":"a405efa4340ed638ddc3db160b8e28b330b8b50e23ef7603990beb25b83f3b63"},"schema_version":"1.0","source":{"id":"1512.01513","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.01513","created_at":"2026-05-18T01:11:17Z"},{"alias_kind":"arxiv_version","alias_value":"1512.01513v2","created_at":"2026-05-18T01:11:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01513","created_at":"2026-05-18T01:11:17Z"},{"alias_kind":"pith_short_12","alias_value":"3KMWANBXEFLS","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"3KMWANBXEFLSDNFC","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"3KMWANBX","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:f490fa44dbb328b6b176e6ba450c8b3a3dd46af0d3d5fe79302b77c4f9f9d70d","target":"graph","created_at":"2026-05-18T01:11:17Z","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":"This work introduces a new kind of semigroup of $\\N^p$ called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We prove they are finitely generated and we give an algorithm to compute their minimal generating sets. We also specialise on the case $p=2$. For this case, we provide a faster algorithm to compute their minimal system of generators and we prove they are Cohen-Macaulay and Buchsbaum. Besides, the Gorenstein property is charactized, and their (minimal) ","authors_text":"A. Vigneron-Tenorio, J. I. Garc\\'ia-Garc\\'ia, M. A. Moreno-Fr\\'ias","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-12-04T19:11:12Z","title":"Proportionally modular affine semigroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01513","kind":"arxiv","version":2},"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:a4290ed292e306f4ab45517a2d3f819b919dbac37cbb0e1dd1d1e0888d3c6b98","target":"record","created_at":"2026-05-18T01:11:17Z","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":"682e2438ed15bbdf0b2ff63dc946e293ffb3a9d6f74e035dd6a05d098e4d02da","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-12-04T19:11:12Z","title_canon_sha256":"a405efa4340ed638ddc3db160b8e28b330b8b50e23ef7603990beb25b83f3b63"},"schema_version":"1.0","source":{"id":"1512.01513","kind":"arxiv","version":2}},"canonical_sha256":"da99603437215721b4a2a79f91c26c191f7db54bc2f914ea78e00dd403b7d27c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da99603437215721b4a2a79f91c26c191f7db54bc2f914ea78e00dd403b7d27c","first_computed_at":"2026-05-18T01:11:17.885428Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:17.885428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u6Sx9C365Oiikg0ViWQ0lMV/IeofZ4wTD3/nVaP00kJJLkeanLyJsWuTu2JSkxhGys1hMoxu0IjMyNdSyKTQCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:17.886100Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.01513","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a4290ed292e306f4ab45517a2d3f819b919dbac37cbb0e1dd1d1e0888d3c6b98","sha256:f490fa44dbb328b6b176e6ba450c8b3a3dd46af0d3d5fe79302b77c4f9f9d70d"],"state_sha256":"b39ba5020c94c7914ddd07d291e5d0adc6c5009b173074a042c65275ef686d1d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PajprZKarga6TUtzsThY5jjq+SwZKM4u7yIl2ZAWHgz26UTyfrljmlnClZ7y14hP26V03cevMevf0qEfrszPDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T01:44:05.070466Z","bundle_sha256":"9dbd5f3452573ceda0fa751a624f80878894ba8219118cd81aeea7b0c75f0a1a"}}