{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:CIJXYDHYPH7S2UBCNO3ACE3SZH","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":"b99a847ca5555659c30b4009c5db16010d6e9568664358be7c3ecb80ed313680","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-06-18T07:42:32Z","title_canon_sha256":"2209c8726090297a7c0046569c988d269a886f9084f56f289eaeafed4b26c4b5"},"schema_version":"1.0","source":{"id":"1806.06531","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.06531","created_at":"2026-05-18T00:13:02Z"},{"alias_kind":"arxiv_version","alias_value":"1806.06531v1","created_at":"2026-05-18T00:13:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.06531","created_at":"2026-05-18T00:13:02Z"},{"alias_kind":"pith_short_12","alias_value":"CIJXYDHYPH7S","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"CIJXYDHYPH7S2UBC","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"CIJXYDHY","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:6808930a02d8157aa5cded63423965c171c131e3c95869906cf3c34388cf574f","target":"graph","created_at":"2026-05-18T00:13:02Z","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 give a direct straightforward proof that there is an isomorphism between the algebra of the Catalan monoid C_n that is, the monoid of all order-preserving, weakly increasing self-maps f of [n] = {1,...,n}, over any commutative ring with identity and the incidence algebra of a certain poset over the ring.","authors_text":"Benjamin Steinberg, Stuart Margolis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-06-18T07:42:32Z","title":"The algebra of the Catalan monoid as an incidence algebra: A simple proof"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06531","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:c3d785de7bdeee36637ff95fd207c7a7f88158f023cfa85acaa45e0506e4e65b","target":"record","created_at":"2026-05-18T00:13:02Z","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":"b99a847ca5555659c30b4009c5db16010d6e9568664358be7c3ecb80ed313680","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-06-18T07:42:32Z","title_canon_sha256":"2209c8726090297a7c0046569c988d269a886f9084f56f289eaeafed4b26c4b5"},"schema_version":"1.0","source":{"id":"1806.06531","kind":"arxiv","version":1}},"canonical_sha256":"12137c0cf879ff2d50226bb6011372c9e56dda743249cc36da69725e02292569","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"12137c0cf879ff2d50226bb6011372c9e56dda743249cc36da69725e02292569","first_computed_at":"2026-05-18T00:13:02.075021Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:02.075021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uXDqqG53fW8MMVEtHAVzo5q5eFI2B5qXfJRnqdw+iWZJfgsBprLq4TVzf5erEdka+C6z3pdge9CyxufbKA79Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:02.075773Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.06531","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3d785de7bdeee36637ff95fd207c7a7f88158f023cfa85acaa45e0506e4e65b","sha256:6808930a02d8157aa5cded63423965c171c131e3c95869906cf3c34388cf574f"],"state_sha256":"c88e57242c8bba744c434d2a949e38fd0e759b967ee39d903e59fc275e5c6037"}