{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:RJHH52TACP4QEVMBMZEPKVIX3P","short_pith_number":"pith:RJHH52TA","canonical_record":{"source":{"id":"1304.4180","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-15T17:26:05Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"d4944bf42ff2036be27272915a7e625c25469665b4c6d6a8a52cf72933216f17","abstract_canon_sha256":"b679c4fce8420ed823d1752656edfaec4bfe6ab6e03fec98a9a7244dc98cf4a0"},"schema_version":"1.0"},"canonical_sha256":"8a4e7eea6013f90255816648f55517dbc9c4fbecdd0a3f6697ff47d67eee80de","source":{"kind":"arxiv","id":"1304.4180","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4180","created_at":"2026-05-18T03:10:36Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4180v1","created_at":"2026-05-18T03:10:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4180","created_at":"2026-05-18T03:10:36Z"},{"alias_kind":"pith_short_12","alias_value":"RJHH52TACP4Q","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"RJHH52TACP4QEVMB","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"RJHH52TA","created_at":"2026-05-18T12:27:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:RJHH52TACP4QEVMBMZEPKVIX3P","target":"record","payload":{"canonical_record":{"source":{"id":"1304.4180","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-15T17:26:05Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"d4944bf42ff2036be27272915a7e625c25469665b4c6d6a8a52cf72933216f17","abstract_canon_sha256":"b679c4fce8420ed823d1752656edfaec4bfe6ab6e03fec98a9a7244dc98cf4a0"},"schema_version":"1.0"},"canonical_sha256":"8a4e7eea6013f90255816648f55517dbc9c4fbecdd0a3f6697ff47d67eee80de","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:36.877042Z","signature_b64":"3Yw8pJt+L8kXzW4DoMV68/31Tr637lqQgnMOPpgpWePakZLhsX/y7U6B9xVQFw+iyR18V8VmonZiG7s/4LlZAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a4e7eea6013f90255816648f55517dbc9c4fbecdd0a3f6697ff47d67eee80de","last_reissued_at":"2026-05-18T03:10:36.876336Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:36.876336Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.4180","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":"2026-05-18T03:10:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5KEOWDA9pkCqhFUAJCFEMueSwYrn27iPr0Tj0KMMu/xoE6WfUXGR17O8IxDzy7Uxiu0mOovxvah4zvt43r5QBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T14:28:23.735910Z"},"content_sha256":"e16a24bb4677bb7b6491c95cfe74c934932c2702aa67b6f747d19ee317c711f0","schema_version":"1.0","event_id":"sha256:e16a24bb4677bb7b6491c95cfe74c934932c2702aa67b6f747d19ee317c711f0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:RJHH52TACP4QEVMBMZEPKVIX3P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The lumpability property for a family of Markov chains on poset block structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Alfredo Donno, Daniele D'Angeli","submitted_at":"2013-04-15T17:26:05Z","abstract_excerpt":"We construct different classes of lumpings for a family of Markov chain products which reflect the structure of a given finite poset. We use essentially combinatorial methods. We prove that, for such a product, every lumping can be obtained from the action of a suitable subgroup of the generalized wreath product of symmetric groups, acting on the underlying poset block structure, if and only if the poset defining the Markov process is totally ordered, and one takes the uniform Markov operator in each factor state space. Finally we show that, when the state space is a homogeneous space associat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4180","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":"2026-05-18T03:10:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GMhCw3j2hvNTAC2e5ownFSp9hqR22x16v3LVwYF5tow0Qvqj891hko/NiStKT4wjZSXelsR2bhP+mQs51VX+AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T14:28:23.736264Z"},"content_sha256":"eadb93723b1b00f0e24ed75a19aa0543a11797b9e9dd6263a5d3017178335e37","schema_version":"1.0","event_id":"sha256:eadb93723b1b00f0e24ed75a19aa0543a11797b9e9dd6263a5d3017178335e37"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RJHH52TACP4QEVMBMZEPKVIX3P/bundle.json","state_url":"https://pith.science/pith/RJHH52TACP4QEVMBMZEPKVIX3P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RJHH52TACP4QEVMBMZEPKVIX3P/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":"2026-06-21T14:28:23Z","links":{"resolver":"https://pith.science/pith/RJHH52TACP4QEVMBMZEPKVIX3P","bundle":"https://pith.science/pith/RJHH52TACP4QEVMBMZEPKVIX3P/bundle.json","state":"https://pith.science/pith/RJHH52TACP4QEVMBMZEPKVIX3P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RJHH52TACP4QEVMBMZEPKVIX3P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:RJHH52TACP4QEVMBMZEPKVIX3P","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":"b679c4fce8420ed823d1752656edfaec4bfe6ab6e03fec98a9a7244dc98cf4a0","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-15T17:26:05Z","title_canon_sha256":"d4944bf42ff2036be27272915a7e625c25469665b4c6d6a8a52cf72933216f17"},"schema_version":"1.0","source":{"id":"1304.4180","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4180","created_at":"2026-05-18T03:10:36Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4180v1","created_at":"2026-05-18T03:10:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4180","created_at":"2026-05-18T03:10:36Z"},{"alias_kind":"pith_short_12","alias_value":"RJHH52TACP4Q","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"RJHH52TACP4QEVMB","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"RJHH52TA","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:eadb93723b1b00f0e24ed75a19aa0543a11797b9e9dd6263a5d3017178335e37","target":"graph","created_at":"2026-05-18T03:10:36Z","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 construct different classes of lumpings for a family of Markov chain products which reflect the structure of a given finite poset. We use essentially combinatorial methods. We prove that, for such a product, every lumping can be obtained from the action of a suitable subgroup of the generalized wreath product of symmetric groups, acting on the underlying poset block structure, if and only if the poset defining the Markov process is totally ordered, and one takes the uniform Markov operator in each factor state space. Finally we show that, when the state space is a homogeneous space associat","authors_text":"Alfredo Donno, Daniele D'Angeli","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-15T17:26:05Z","title":"The lumpability property for a family of Markov chains on poset block structures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4180","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:e16a24bb4677bb7b6491c95cfe74c934932c2702aa67b6f747d19ee317c711f0","target":"record","created_at":"2026-05-18T03:10:36Z","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":"b679c4fce8420ed823d1752656edfaec4bfe6ab6e03fec98a9a7244dc98cf4a0","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-15T17:26:05Z","title_canon_sha256":"d4944bf42ff2036be27272915a7e625c25469665b4c6d6a8a52cf72933216f17"},"schema_version":"1.0","source":{"id":"1304.4180","kind":"arxiv","version":1}},"canonical_sha256":"8a4e7eea6013f90255816648f55517dbc9c4fbecdd0a3f6697ff47d67eee80de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8a4e7eea6013f90255816648f55517dbc9c4fbecdd0a3f6697ff47d67eee80de","first_computed_at":"2026-05-18T03:10:36.876336Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:10:36.876336Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3Yw8pJt+L8kXzW4DoMV68/31Tr637lqQgnMOPpgpWePakZLhsX/y7U6B9xVQFw+iyR18V8VmonZiG7s/4LlZAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:10:36.877042Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.4180","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e16a24bb4677bb7b6491c95cfe74c934932c2702aa67b6f747d19ee317c711f0","sha256:eadb93723b1b00f0e24ed75a19aa0543a11797b9e9dd6263a5d3017178335e37"],"state_sha256":"79dee7b096eeb43b202cf231a7a19c26c2e6d6be984f2891d6ecd78ceb6b235e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H5M+M8QDe/wPip+pV3vSD33DX+khOOXOwZmPvEFnBWgejsxz7omaD4rjAaht6u71GsnsoUMNUWezV4OfukiOAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T14:28:23.738262Z","bundle_sha256":"a5e99841831fdc407370c5977c655ef2d5dcda1a9ece32c5edddb451273454df"}}