{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:FP27M4AXPRP6MVJAATKV6FUH7Z","short_pith_number":"pith:FP27M4AX","canonical_record":{"source":{"id":"1610.04122","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-10-13T15:17:58Z","cross_cats_sorted":[],"title_canon_sha256":"c21225d5a83f16d9669269b45f348365ed817b8403988aacc73ce8cbc7518b4d","abstract_canon_sha256":"76e126def7b2f711d201da0c1a74d93f6945feefce6d2dd8dd607a74fce3861a"},"schema_version":"1.0"},"canonical_sha256":"2bf5f670177c5fe6552004d55f1687fe5d195cdf74de05f3ecc07117728f987e","source":{"kind":"arxiv","id":"1610.04122","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.04122","created_at":"2026-05-18T01:02:21Z"},{"alias_kind":"arxiv_version","alias_value":"1610.04122v1","created_at":"2026-05-18T01:02:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04122","created_at":"2026-05-18T01:02:21Z"},{"alias_kind":"pith_short_12","alias_value":"FP27M4AXPRP6","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FP27M4AXPRP6MVJA","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FP27M4AX","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:FP27M4AXPRP6MVJAATKV6FUH7Z","target":"record","payload":{"canonical_record":{"source":{"id":"1610.04122","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-10-13T15:17:58Z","cross_cats_sorted":[],"title_canon_sha256":"c21225d5a83f16d9669269b45f348365ed817b8403988aacc73ce8cbc7518b4d","abstract_canon_sha256":"76e126def7b2f711d201da0c1a74d93f6945feefce6d2dd8dd607a74fce3861a"},"schema_version":"1.0"},"canonical_sha256":"2bf5f670177c5fe6552004d55f1687fe5d195cdf74de05f3ecc07117728f987e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:21.880460Z","signature_b64":"+SI8+6nhnxbPRXc8YiXfABZL9q2ggjmUHHlnDVxDS8FNYwxy0RYkcUPB57N4yUGZXP1VjaWi0bYeKsr19cZPDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2bf5f670177c5fe6552004d55f1687fe5d195cdf74de05f3ecc07117728f987e","last_reissued_at":"2026-05-18T01:02:21.879758Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:21.879758Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.04122","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":"2026-05-18T01:02:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MkpdMzm8dZLPVbhYNdvqQTJwh8U3nF91MxdCWQsV9M7KI32qEy7DRvrz4FL+cF4UpUwbvKS6Q7MuRy+mzf90Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T19:58:49.242271Z"},"content_sha256":"84460deace9a3e3d34e1d8a3cf7d7ad13758f48fbfdd03c13b8c3a57ccd6f923","schema_version":"1.0","event_id":"sha256:84460deace9a3e3d34e1d8a3cf7d7ad13758f48fbfdd03c13b8c3a57ccd6f923"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:FP27M4AXPRP6MVJAATKV6FUH7Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modules and Structures of Planar Upper Triangular Rook Monoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Jianqiang Feng, Wenli Liu, Ximei Bai, Zhenheng Li","submitted_at":"2016-10-13T15:17:58Z","abstract_excerpt":"In this paper, we discuss modules and structures of the planar upper triangular rook monoid B_n. We first show that the order of B_n is a Catalan number, then we investigate the properties of a module V over B_n generated by a set of elements v_S indexed by the power set of {1, ..., n}. We find that every nonzero submodule of V is cyclic and completely decomposable; we give a necessary and sufficient condition for a submodule of V to be indecomposable. We show that every irreducible submodule of V is 1-dimensional. Furthermore, we give a formula for calculating the dimension of every submodule"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04122","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":"2026-05-18T01:02:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m+iIUqq1WQcXkOSyFbQx8wzHqhkMLqkmTh56CAYGvDRd70yoOk7RqmVdukW7jzM6+kuv+gUrSNw1QbJoH9VnDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T19:58:49.242985Z"},"content_sha256":"f5e5dd56dd96a0864d132701eb39b45437a8081787dcbb3ebc3e33c35b685b2f","schema_version":"1.0","event_id":"sha256:f5e5dd56dd96a0864d132701eb39b45437a8081787dcbb3ebc3e33c35b685b2f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/bundle.json","state_url":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/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":"2026-05-26T19:58:49Z","links":{"resolver":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z","bundle":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/bundle.json","state":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FP27M4AXPRP6MVJAATKV6FUH7Z","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":"76e126def7b2f711d201da0c1a74d93f6945feefce6d2dd8dd607a74fce3861a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-10-13T15:17:58Z","title_canon_sha256":"c21225d5a83f16d9669269b45f348365ed817b8403988aacc73ce8cbc7518b4d"},"schema_version":"1.0","source":{"id":"1610.04122","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.04122","created_at":"2026-05-18T01:02:21Z"},{"alias_kind":"arxiv_version","alias_value":"1610.04122v1","created_at":"2026-05-18T01:02:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04122","created_at":"2026-05-18T01:02:21Z"},{"alias_kind":"pith_short_12","alias_value":"FP27M4AXPRP6","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FP27M4AXPRP6MVJA","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FP27M4AX","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:f5e5dd56dd96a0864d132701eb39b45437a8081787dcbb3ebc3e33c35b685b2f","target":"graph","created_at":"2026-05-18T01:02:21Z","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":"In this paper, we discuss modules and structures of the planar upper triangular rook monoid B_n. We first show that the order of B_n is a Catalan number, then we investigate the properties of a module V over B_n generated by a set of elements v_S indexed by the power set of {1, ..., n}. We find that every nonzero submodule of V is cyclic and completely decomposable; we give a necessary and sufficient condition for a submodule of V to be indecomposable. We show that every irreducible submodule of V is 1-dimensional. Furthermore, we give a formula for calculating the dimension of every submodule","authors_text":"Jianqiang Feng, Wenli Liu, Ximei Bai, Zhenheng Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-10-13T15:17:58Z","title":"Modules and Structures of Planar Upper Triangular Rook Monoids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04122","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":"sha256:84460deace9a3e3d34e1d8a3cf7d7ad13758f48fbfdd03c13b8c3a57ccd6f923","target":"record","created_at":"2026-05-18T01:02:21Z","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":"76e126def7b2f711d201da0c1a74d93f6945feefce6d2dd8dd607a74fce3861a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-10-13T15:17:58Z","title_canon_sha256":"c21225d5a83f16d9669269b45f348365ed817b8403988aacc73ce8cbc7518b4d"},"schema_version":"1.0","source":{"id":"1610.04122","kind":"arxiv","version":1}},"canonical_sha256":"2bf5f670177c5fe6552004d55f1687fe5d195cdf74de05f3ecc07117728f987e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2bf5f670177c5fe6552004d55f1687fe5d195cdf74de05f3ecc07117728f987e","first_computed_at":"2026-05-18T01:02:21.879758Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:02:21.879758Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+SI8+6nhnxbPRXc8YiXfABZL9q2ggjmUHHlnDVxDS8FNYwxy0RYkcUPB57N4yUGZXP1VjaWi0bYeKsr19cZPDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:02:21.880460Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.04122","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84460deace9a3e3d34e1d8a3cf7d7ad13758f48fbfdd03c13b8c3a57ccd6f923","sha256:f5e5dd56dd96a0864d132701eb39b45437a8081787dcbb3ebc3e33c35b685b2f"],"state_sha256":"51fdc137f00c31fb33fc22ebfedca33a039313226eb8c237ea1f0ade169fdd02"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G5dqspLppF11tkikZXCYECgyiqPuc7qjdcv2jM0TfCxurb9uSzfVd8TgekYMTsZ2Cz5XHcQzX8jMGYy0oCkxBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T19:58:49.246651Z","bundle_sha256":"cbd22374051d48f566d9acb7a33554159eb9b9cc2b1513b425e8cf1fef0eb76f"}}