{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:FP27M4AXPRP6MVJAATKV6FUH7Z","short_pith_number":"pith:FP27M4AX","schema_version":"1.0","canonical_sha256":"2bf5f670177c5fe6552004d55f1687fe5d195cdf74de05f3ecc07117728f987e","source":{"kind":"arxiv","id":"1610.04122","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1610.04122","created_at":"2026-05-18T01:02:21.879860+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.04122v1","created_at":"2026-05-18T01:02:21.879860+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04122","created_at":"2026-05-18T01:02:21.879860+00:00"},{"alias_kind":"pith_short_12","alias_value":"FP27M4AXPRP6","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"FP27M4AXPRP6MVJA","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"FP27M4AX","created_at":"2026-05-18T12:30:15.759754+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z","json":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z.json","graph_json":"https://pith.science/api/pith-number/FP27M4AXPRP6MVJAATKV6FUH7Z/graph.json","events_json":"https://pith.science/api/pith-number/FP27M4AXPRP6MVJAATKV6FUH7Z/events.json","paper":"https://pith.science/paper/FP27M4AX"},"agent_actions":{"view_html":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z","download_json":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z.json","view_paper":"https://pith.science/paper/FP27M4AX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.04122&json=true","fetch_graph":"https://pith.science/api/pith-number/FP27M4AXPRP6MVJAATKV6FUH7Z/graph.json","fetch_events":"https://pith.science/api/pith-number/FP27M4AXPRP6MVJAATKV6FUH7Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/action/storage_attestation","attest_author":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/action/author_attestation","sign_citation":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/action/citation_signature","submit_replication":"https://pith.science/pith/FP27M4AXPRP6MVJAATKV6FUH7Z/action/replication_record"}},"created_at":"2026-05-18T01:02:21.879860+00:00","updated_at":"2026-05-18T01:02:21.879860+00:00"}