{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:3Y25GG4VFRNHIDFURPLQKQ3QVD","short_pith_number":"pith:3Y25GG4V","canonical_record":{"source":{"id":"1703.05488","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-03-16T06:55:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d09da81b79aafe0e6d7b3c62740e4c8a8a255e16f22f82a52eb5ecba782e6077","abstract_canon_sha256":"1bc80b103a5aacc08b40a8642afd271f7cf8ba553bf49d05dbeca6e0d0a6e5fa"},"schema_version":"1.0"},"canonical_sha256":"de35d31b952c5a740cb48bd7054370a8d297a143bfb0c550520243a3a7688327","source":{"kind":"arxiv","id":"1703.05488","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.05488","created_at":"2026-05-18T00:48:34Z"},{"alias_kind":"arxiv_version","alias_value":"1703.05488v1","created_at":"2026-05-18T00:48:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.05488","created_at":"2026-05-18T00:48:34Z"},{"alias_kind":"pith_short_12","alias_value":"3Y25GG4VFRNH","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3Y25GG4VFRNHIDFU","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3Y25GG4V","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:3Y25GG4VFRNHIDFURPLQKQ3QVD","target":"record","payload":{"canonical_record":{"source":{"id":"1703.05488","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-03-16T06:55:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d09da81b79aafe0e6d7b3c62740e4c8a8a255e16f22f82a52eb5ecba782e6077","abstract_canon_sha256":"1bc80b103a5aacc08b40a8642afd271f7cf8ba553bf49d05dbeca6e0d0a6e5fa"},"schema_version":"1.0"},"canonical_sha256":"de35d31b952c5a740cb48bd7054370a8d297a143bfb0c550520243a3a7688327","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:34.154675Z","signature_b64":"QGBl/s6i6dScfCYVvvnvjm4hE6bmg6h+DexZLDCxnynYQOwlDp1oMXhAAmpG1MOHxLy6f2rdjeJ4glq1FLFgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de35d31b952c5a740cb48bd7054370a8d297a143bfb0c550520243a3a7688327","last_reissued_at":"2026-05-18T00:48:34.154035Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:34.154035Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.05488","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:48:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gSCJ3lK5/+ELuZYced8WFbYXPVvR2E+AfB1ibl7fOYlopASqIqlr2jQuT+y0pDJjfaXNw8ZuSfJ/26mjBcC9Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:12:05.009812Z"},"content_sha256":"bfee9f8bd1a8c25453c98e4cd84fc2c92293daadb7378b8c35835775f28f4f49","schema_version":"1.0","event_id":"sha256:bfee9f8bd1a8c25453c98e4cd84fc2c92293daadb7378b8c35835775f28f4f49"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:3Y25GG4VFRNHIDFURPLQKQ3QVD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Rahim Rahmati-Asghar","submitted_at":"2017-03-16T06:55:29Z","abstract_excerpt":"We introduce pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes, respectively, as the extensions of the notions of $k$-clean monomial ideals and $k$-decomposable simplicial complexes. We show that a multicomplex $\\Gamma$ is $k$-decomposable if and only if its associated monomial ideal $I(\\Gamma)$ is pretty $k$-clean. Also, we prove that an arbitrary monomial ideal $I$ is pretty $k$-clean if and only if its polarization $I^p$ is $k$-clean. Our results extend and generalize some results due to Herzog-Popescu, Soleyman Jahan and the current author."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05488","kind":"arxiv","version":1},"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":"2026-05-18T00:48:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Fqz291BI7Q3/rcsfLH4IkkFVSR6DdA3Z7BAN2WADwcDd99RaVRgAiJh51TNEuh6zD3ZQl6evIXxPe87UdwGAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:12:05.010157Z"},"content_sha256":"0a4f69176b9a18bce35199c9387157a285397f03d204f86d46623dd9c1af1106","schema_version":"1.0","event_id":"sha256:0a4f69176b9a18bce35199c9387157a285397f03d204f86d46623dd9c1af1106"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3Y25GG4VFRNHIDFURPLQKQ3QVD/bundle.json","state_url":"https://pith.science/pith/3Y25GG4VFRNHIDFURPLQKQ3QVD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3Y25GG4VFRNHIDFURPLQKQ3QVD/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":"2026-06-06T20:12:05Z","links":{"resolver":"https://pith.science/pith/3Y25GG4VFRNHIDFURPLQKQ3QVD","bundle":"https://pith.science/pith/3Y25GG4VFRNHIDFURPLQKQ3QVD/bundle.json","state":"https://pith.science/pith/3Y25GG4VFRNHIDFURPLQKQ3QVD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3Y25GG4VFRNHIDFURPLQKQ3QVD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:3Y25GG4VFRNHIDFURPLQKQ3QVD","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":"1bc80b103a5aacc08b40a8642afd271f7cf8ba553bf49d05dbeca6e0d0a6e5fa","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-03-16T06:55:29Z","title_canon_sha256":"d09da81b79aafe0e6d7b3c62740e4c8a8a255e16f22f82a52eb5ecba782e6077"},"schema_version":"1.0","source":{"id":"1703.05488","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.05488","created_at":"2026-05-18T00:48:34Z"},{"alias_kind":"arxiv_version","alias_value":"1703.05488v1","created_at":"2026-05-18T00:48:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.05488","created_at":"2026-05-18T00:48:34Z"},{"alias_kind":"pith_short_12","alias_value":"3Y25GG4VFRNH","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3Y25GG4VFRNHIDFU","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3Y25GG4V","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:0a4f69176b9a18bce35199c9387157a285397f03d204f86d46623dd9c1af1106","target":"graph","created_at":"2026-05-18T00:48:34Z","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":"We introduce pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes, respectively, as the extensions of the notions of $k$-clean monomial ideals and $k$-decomposable simplicial complexes. We show that a multicomplex $\\Gamma$ is $k$-decomposable if and only if its associated monomial ideal $I(\\Gamma)$ is pretty $k$-clean. Also, we prove that an arbitrary monomial ideal $I$ is pretty $k$-clean if and only if its polarization $I^p$ is $k$-clean. Our results extend and generalize some results due to Herzog-Popescu, Soleyman Jahan and the current author.","authors_text":"Rahim Rahmati-Asghar","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-03-16T06:55:29Z","title":"Pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05488","kind":"arxiv","version":1},"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:bfee9f8bd1a8c25453c98e4cd84fc2c92293daadb7378b8c35835775f28f4f49","target":"record","created_at":"2026-05-18T00:48:34Z","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":"1bc80b103a5aacc08b40a8642afd271f7cf8ba553bf49d05dbeca6e0d0a6e5fa","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-03-16T06:55:29Z","title_canon_sha256":"d09da81b79aafe0e6d7b3c62740e4c8a8a255e16f22f82a52eb5ecba782e6077"},"schema_version":"1.0","source":{"id":"1703.05488","kind":"arxiv","version":1}},"canonical_sha256":"de35d31b952c5a740cb48bd7054370a8d297a143bfb0c550520243a3a7688327","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"de35d31b952c5a740cb48bd7054370a8d297a143bfb0c550520243a3a7688327","first_computed_at":"2026-05-18T00:48:34.154035Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:48:34.154035Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QGBl/s6i6dScfCYVvvnvjm4hE6bmg6h+DexZLDCxnynYQOwlDp1oMXhAAmpG1MOHxLy6f2rdjeJ4glq1FLFgDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:48:34.154675Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.05488","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bfee9f8bd1a8c25453c98e4cd84fc2c92293daadb7378b8c35835775f28f4f49","sha256:0a4f69176b9a18bce35199c9387157a285397f03d204f86d46623dd9c1af1106"],"state_sha256":"979221b7bf982420e50ed68ea249560bde1c87e21a022c6288690153ca80ac45"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iGooJkE4QcgavDISIYZB2vsM5Hn06O9ekxEvZJ/6yCMa1exsRg4roiXpt8dMtisvNHcLOvWqYNDudOryYvWIAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T20:12:05.012130Z","bundle_sha256":"6826ec14e99bef3fc55004e1d3fdd6a44bbaebba119afeb4aa1686019d73b612"}}