{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:XD647BPMKJTEZZL5LY2SZRVDZP","short_pith_number":"pith:XD647BPM","canonical_record":{"source":{"id":"1504.01789","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-08T00:18:00Z","cross_cats_sorted":["cs.LO"],"title_canon_sha256":"7ed41f0a9406a3b6c7485255d0b488f6090d4503fe626696f06370b06ee76399","abstract_canon_sha256":"17241666c088366d958ff06e4d19b083386bd6a41ea26eabf525518fa78b77ec"},"schema_version":"1.0"},"canonical_sha256":"b8fdcf85ec52664ce57d5e352cc6a3cbf7a50d265ea0b28f10eb81512ddf491f","source":{"kind":"arxiv","id":"1504.01789","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.01789","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"arxiv_version","alias_value":"1504.01789v2","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.01789","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"pith_short_12","alias_value":"XD647BPMKJTE","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XD647BPMKJTEZZL5","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XD647BPM","created_at":"2026-05-18T12:29:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:XD647BPMKJTEZZL5LY2SZRVDZP","target":"record","payload":{"canonical_record":{"source":{"id":"1504.01789","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-08T00:18:00Z","cross_cats_sorted":["cs.LO"],"title_canon_sha256":"7ed41f0a9406a3b6c7485255d0b488f6090d4503fe626696f06370b06ee76399","abstract_canon_sha256":"17241666c088366d958ff06e4d19b083386bd6a41ea26eabf525518fa78b77ec"},"schema_version":"1.0"},"canonical_sha256":"b8fdcf85ec52664ce57d5e352cc6a3cbf7a50d265ea0b28f10eb81512ddf491f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:44.144105Z","signature_b64":"s9iHkt42Z4g8qhAZ5/aE4tQWNBYpzV3wJX5p29ahB+JlMKYRYkLb3Nd2qx+3NLsuvips7z6WhYUMVaqgLNNaCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8fdcf85ec52664ce57d5e352cc6a3cbf7a50d265ea0b28f10eb81512ddf491f","last_reissued_at":"2026-05-18T00:46:44.143457Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:44.143457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.01789","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:46:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1ToR+u8Wuzifh9VghldxBUe9Gm2Njp9aXqFhU7yuXH1q0cGQAg4OQBpt2wsxJnP85aBYUalASeE/3ZhQ2VsZAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T08:56:47.232572Z"},"content_sha256":"8a8e073237114598cb4d2052aacb6599a3c82f07c30d2ae8ba357ae669d7cf22","schema_version":"1.0","event_id":"sha256:8a8e073237114598cb4d2052aacb6599a3c82f07c30d2ae8ba357ae669d7cf22"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:XD647BPMKJTEZZL5LY2SZRVDZP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Lattice of Congruences of a Finite Line Frame","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LO"],"primary_cat":"math.LO","authors_text":"Carlos Areces, Daniel Penazzi, Miguel Campercholi, Pedro S\\'anchez Terraf","submitted_at":"2015-04-08T00:18:00Z","abstract_excerpt":"Let $\\mathbf{F}=\\left\\langle F,R\\right\\rangle $ be a finite Kripke frame. A congruence of $\\mathbf{F}$ is a bisimulation of $\\mathbf{F}$ that is also an equivalence relation on F. The set of all congruences of $\\mathbf{F}$ is a lattice under the inclusion ordering. In this article we investigate this lattice in the case that $\\mathbf{F}$ is a finite line frame. We give concrete descriptions of the join and meet of two congruences with a nontrivial upper bound. Through these descriptions we show that for every nontrivial congruence $\\rho$, the interval $[\\mathrm{Id_{F},\\rho]}$ embeds into the l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01789","kind":"arxiv","version":2},"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":"2026-05-18T00:46:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QglOc4W9o7nkh77gswqTnt8pAZuyiCOfaGMeuP6l5b3rp/Ovmv5Htkm1ouHOMk4lh8GC5pnwBlKTQegNTJjQCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T08:56:47.232938Z"},"content_sha256":"5a27ae7704917501cdd04afc92cd2054a19898cdf4e732bf9386a0210635a220","schema_version":"1.0","event_id":"sha256:5a27ae7704917501cdd04afc92cd2054a19898cdf4e732bf9386a0210635a220"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XD647BPMKJTEZZL5LY2SZRVDZP/bundle.json","state_url":"https://pith.science/pith/XD647BPMKJTEZZL5LY2SZRVDZP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XD647BPMKJTEZZL5LY2SZRVDZP/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":"2026-06-27T08:56:47Z","links":{"resolver":"https://pith.science/pith/XD647BPMKJTEZZL5LY2SZRVDZP","bundle":"https://pith.science/pith/XD647BPMKJTEZZL5LY2SZRVDZP/bundle.json","state":"https://pith.science/pith/XD647BPMKJTEZZL5LY2SZRVDZP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XD647BPMKJTEZZL5LY2SZRVDZP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:XD647BPMKJTEZZL5LY2SZRVDZP","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":"17241666c088366d958ff06e4d19b083386bd6a41ea26eabf525518fa78b77ec","cross_cats_sorted":["cs.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-08T00:18:00Z","title_canon_sha256":"7ed41f0a9406a3b6c7485255d0b488f6090d4503fe626696f06370b06ee76399"},"schema_version":"1.0","source":{"id":"1504.01789","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.01789","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"arxiv_version","alias_value":"1504.01789v2","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.01789","created_at":"2026-05-18T00:46:44Z"},{"alias_kind":"pith_short_12","alias_value":"XD647BPMKJTE","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XD647BPMKJTEZZL5","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XD647BPM","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:5a27ae7704917501cdd04afc92cd2054a19898cdf4e732bf9386a0210635a220","target":"graph","created_at":"2026-05-18T00:46:44Z","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":"Let $\\mathbf{F}=\\left\\langle F,R\\right\\rangle $ be a finite Kripke frame. A congruence of $\\mathbf{F}$ is a bisimulation of $\\mathbf{F}$ that is also an equivalence relation on F. The set of all congruences of $\\mathbf{F}$ is a lattice under the inclusion ordering. In this article we investigate this lattice in the case that $\\mathbf{F}$ is a finite line frame. We give concrete descriptions of the join and meet of two congruences with a nontrivial upper bound. Through these descriptions we show that for every nontrivial congruence $\\rho$, the interval $[\\mathrm{Id_{F},\\rho]}$ embeds into the l","authors_text":"Carlos Areces, Daniel Penazzi, Miguel Campercholi, Pedro S\\'anchez Terraf","cross_cats":["cs.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-08T00:18:00Z","title":"The Lattice of Congruences of a Finite Line Frame"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01789","kind":"arxiv","version":2},"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":"sha256:8a8e073237114598cb4d2052aacb6599a3c82f07c30d2ae8ba357ae669d7cf22","target":"record","created_at":"2026-05-18T00:46:44Z","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":"17241666c088366d958ff06e4d19b083386bd6a41ea26eabf525518fa78b77ec","cross_cats_sorted":["cs.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-04-08T00:18:00Z","title_canon_sha256":"7ed41f0a9406a3b6c7485255d0b488f6090d4503fe626696f06370b06ee76399"},"schema_version":"1.0","source":{"id":"1504.01789","kind":"arxiv","version":2}},"canonical_sha256":"b8fdcf85ec52664ce57d5e352cc6a3cbf7a50d265ea0b28f10eb81512ddf491f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b8fdcf85ec52664ce57d5e352cc6a3cbf7a50d265ea0b28f10eb81512ddf491f","first_computed_at":"2026-05-18T00:46:44.143457Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:44.143457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s9iHkt42Z4g8qhAZ5/aE4tQWNBYpzV3wJX5p29ahB+JlMKYRYkLb3Nd2qx+3NLsuvips7z6WhYUMVaqgLNNaCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:44.144105Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.01789","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a8e073237114598cb4d2052aacb6599a3c82f07c30d2ae8ba357ae669d7cf22","sha256:5a27ae7704917501cdd04afc92cd2054a19898cdf4e732bf9386a0210635a220"],"state_sha256":"38cc45e4f6cd696adcb928dacbbc3c6ad00e3f0a7f77b10b6523099932b31ac5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yEvzZENo8mqbVuc7pldHnCotcl2HCY/+P89rk2URx4rGTm/NCKCGPcfEYJMhR8np5eVP37F1jPM3Cx/yo0KZAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T08:56:47.234845Z","bundle_sha256":"5662b264b4b3d9ec3eba0ff720166c2fd4f27c7f08bbdbc2928af19a6b8f7b7e"}}