{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:4DQEITOZAAITRUIYHADCX3QVIU","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":"d5eeb3fffed4de43fd05dc6625c254d15a6d824de94f4acf1d4249321da299c7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-04-30T18:53:18Z","title_canon_sha256":"74db557c08bb12f552fe5710ead06d6147d740184d97bb06aa8cbf9e68fe8b5c"},"schema_version":"1.0","source":{"id":"1004.5589","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.5589","created_at":"2026-05-18T02:24:09Z"},{"alias_kind":"arxiv_version","alias_value":"1004.5589v1","created_at":"2026-05-18T02:24:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5589","created_at":"2026-05-18T02:24:09Z"},{"alias_kind":"pith_short_12","alias_value":"4DQEITOZAAIT","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"4DQEITOZAAITRUIY","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"4DQEITOZ","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:7ffed3bbcae6805c4a2a524e69eb305c0be6f4fe8f317405b031ff2f30bfeab0","target":"graph","created_at":"2026-05-18T02:24:09Z","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":"The Bernoulli measure on strings is used to define height functions for the dense R- and L-orders of the Thompson-Higman monoids M_{k,1}. The measure can also be used to characterize the D-relation of certain submonoids of M_{k,1}. The computational complexity of computing the Bernoulli measure of certain sets, and in particular, of computing the R- and L-height of an element of M_{k,1} is investigated.","authors_text":"Jean-Camille Birget","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-04-30T18:53:18Z","title":"Bernoulli measure on strings, and Thompson-Higman monoids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5589","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:d645adb1e44440514137ce9650d8f8f542af4268cb78bd491bb866176ea6c8af","target":"record","created_at":"2026-05-18T02:24:09Z","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":"d5eeb3fffed4de43fd05dc6625c254d15a6d824de94f4acf1d4249321da299c7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-04-30T18:53:18Z","title_canon_sha256":"74db557c08bb12f552fe5710ead06d6147d740184d97bb06aa8cbf9e68fe8b5c"},"schema_version":"1.0","source":{"id":"1004.5589","kind":"arxiv","version":1}},"canonical_sha256":"e0e0444dd9001138d11838062bee1545112ed2b32cc4c9d3f3ec3d478a3c73fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e0e0444dd9001138d11838062bee1545112ed2b32cc4c9d3f3ec3d478a3c73fa","first_computed_at":"2026-05-18T02:24:09.749986Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:09.749986Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hK00GiyFl5hKR4xpJ7a5LxhAZe3pXL6ULgSsv3lt6uZvBHgkyTxZVyEZq3eGDeAR+zwf2Mix/3b/0g9j3ArCCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:09.750647Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.5589","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d645adb1e44440514137ce9650d8f8f542af4268cb78bd491bb866176ea6c8af","sha256:7ffed3bbcae6805c4a2a524e69eb305c0be6f4fe8f317405b031ff2f30bfeab0"],"state_sha256":"401a7316b06615c3967a3b5ced7c85a371a3f373454587625e5fce63efed20cc"}