{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:XXCJWJL22APGH5FNAIVK66G2IA","short_pith_number":"pith:XXCJWJL2","canonical_record":{"source":{"id":"1008.0170","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CL","submitted_at":"2010-08-01T12:45:58Z","cross_cats_sorted":[],"title_canon_sha256":"42848c79676d6aaefd507621cb8a390a3ecb59de2a7f9c7dcf39e93798455f68","abstract_canon_sha256":"e70c098ed4667548db2befa8b2f898f6b0ec6ea60de1ed329b5f4da163af1e16"},"schema_version":"1.0"},"canonical_sha256":"bdc49b257ad01e63f4ad022aaf78da400bfc4e30c876b6546fd6b7b877796378","source":{"kind":"arxiv","id":"1008.0170","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.0170","created_at":"2026-05-18T04:42:43Z"},{"alias_kind":"arxiv_version","alias_value":"1008.0170v1","created_at":"2026-05-18T04:42:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.0170","created_at":"2026-05-18T04:42:43Z"},{"alias_kind":"pith_short_12","alias_value":"XXCJWJL22APG","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XXCJWJL22APGH5FN","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XXCJWJL2","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:XXCJWJL22APGH5FNAIVK66G2IA","target":"record","payload":{"canonical_record":{"source":{"id":"1008.0170","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CL","submitted_at":"2010-08-01T12:45:58Z","cross_cats_sorted":[],"title_canon_sha256":"42848c79676d6aaefd507621cb8a390a3ecb59de2a7f9c7dcf39e93798455f68","abstract_canon_sha256":"e70c098ed4667548db2befa8b2f898f6b0ec6ea60de1ed329b5f4da163af1e16"},"schema_version":"1.0"},"canonical_sha256":"bdc49b257ad01e63f4ad022aaf78da400bfc4e30c876b6546fd6b7b877796378","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:43.901079Z","signature_b64":"Iuj06eVt/zAu7MIxhG5Hfayd0WaNlR6ASJ9wkKuCDMfW26BSAtE1jiZAei0t5V5+oFHZFb7Odoj2605+EyfLDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdc49b257ad01e63f4ad022aaf78da400bfc4e30c876b6546fd6b7b877796378","last_reissued_at":"2026-05-18T04:42:43.900420Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:43.900420Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.0170","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":"2026-05-18T04:42:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mJTrRsS7qIW1sBT/Se0xIHcqSq1KZ0qlJKo93wB5gQ9TwHa45hdWKPA6/60bh5NO2nncKj6WzdR8XVshxO71AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T21:49:05.790050Z"},"content_sha256":"ba875d6bb316811e27355367aa26a321d6d9076fd7677d284e0924670e0fa369","schema_version":"1.0","event_id":"sha256:ba875d6bb316811e27355367aa26a321d6d9076fd7677d284e0924670e0fa369"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:XXCJWJL22APGH5FNAIVK66G2IA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Symmetric categorial grammar: residuation and Galois connections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CL","authors_text":"Michael Moortgat","submitted_at":"2010-08-01T12:45:58Z","abstract_excerpt":"The Lambek-Grishin calculus is a symmetric extension of the Lambek calculus: in addition to the residuated family of product, left and right division operations of Lambek's original calculus, one also considers a family of coproduct, right and left difference operations, related to the former by an arrow-reversing duality. Communication between the two families is implemented in terms of linear distributivity principles. The aim of this paper is to complement the symmetry between (dual) residuated type-forming operations with an orthogonal opposition that contrasts residuated and Galois connec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.0170","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":"2026-05-18T04:42:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MKy9PqCOrJDNuetCsGyQszIM2IuoSPtIRlTuPbgEJOsyf/tRQ5w7b3SBtU5+VP/gzEmCbd+o2IhIM1hIPqfxAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T21:49:05.790696Z"},"content_sha256":"64865480c4690355578843a0309b2e65eb6d0f2a6a0e2ae8f7ef984981bd3f03","schema_version":"1.0","event_id":"sha256:64865480c4690355578843a0309b2e65eb6d0f2a6a0e2ae8f7ef984981bd3f03"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XXCJWJL22APGH5FNAIVK66G2IA/bundle.json","state_url":"https://pith.science/pith/XXCJWJL22APGH5FNAIVK66G2IA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XXCJWJL22APGH5FNAIVK66G2IA/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":"2026-05-28T21:49:05Z","links":{"resolver":"https://pith.science/pith/XXCJWJL22APGH5FNAIVK66G2IA","bundle":"https://pith.science/pith/XXCJWJL22APGH5FNAIVK66G2IA/bundle.json","state":"https://pith.science/pith/XXCJWJL22APGH5FNAIVK66G2IA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XXCJWJL22APGH5FNAIVK66G2IA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:XXCJWJL22APGH5FNAIVK66G2IA","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":"e70c098ed4667548db2befa8b2f898f6b0ec6ea60de1ed329b5f4da163af1e16","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CL","submitted_at":"2010-08-01T12:45:58Z","title_canon_sha256":"42848c79676d6aaefd507621cb8a390a3ecb59de2a7f9c7dcf39e93798455f68"},"schema_version":"1.0","source":{"id":"1008.0170","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.0170","created_at":"2026-05-18T04:42:43Z"},{"alias_kind":"arxiv_version","alias_value":"1008.0170v1","created_at":"2026-05-18T04:42:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.0170","created_at":"2026-05-18T04:42:43Z"},{"alias_kind":"pith_short_12","alias_value":"XXCJWJL22APG","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XXCJWJL22APGH5FN","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XXCJWJL2","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:64865480c4690355578843a0309b2e65eb6d0f2a6a0e2ae8f7ef984981bd3f03","target":"graph","created_at":"2026-05-18T04:42:43Z","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 Lambek-Grishin calculus is a symmetric extension of the Lambek calculus: in addition to the residuated family of product, left and right division operations of Lambek's original calculus, one also considers a family of coproduct, right and left difference operations, related to the former by an arrow-reversing duality. Communication between the two families is implemented in terms of linear distributivity principles. The aim of this paper is to complement the symmetry between (dual) residuated type-forming operations with an orthogonal opposition that contrasts residuated and Galois connec","authors_text":"Michael Moortgat","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CL","submitted_at":"2010-08-01T12:45:58Z","title":"Symmetric categorial grammar: residuation and Galois connections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.0170","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:ba875d6bb316811e27355367aa26a321d6d9076fd7677d284e0924670e0fa369","target":"record","created_at":"2026-05-18T04:42:43Z","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":"e70c098ed4667548db2befa8b2f898f6b0ec6ea60de1ed329b5f4da163af1e16","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CL","submitted_at":"2010-08-01T12:45:58Z","title_canon_sha256":"42848c79676d6aaefd507621cb8a390a3ecb59de2a7f9c7dcf39e93798455f68"},"schema_version":"1.0","source":{"id":"1008.0170","kind":"arxiv","version":1}},"canonical_sha256":"bdc49b257ad01e63f4ad022aaf78da400bfc4e30c876b6546fd6b7b877796378","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bdc49b257ad01e63f4ad022aaf78da400bfc4e30c876b6546fd6b7b877796378","first_computed_at":"2026-05-18T04:42:43.900420Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:43.900420Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Iuj06eVt/zAu7MIxhG5Hfayd0WaNlR6ASJ9wkKuCDMfW26BSAtE1jiZAei0t5V5+oFHZFb7Odoj2605+EyfLDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:43.901079Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.0170","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba875d6bb316811e27355367aa26a321d6d9076fd7677d284e0924670e0fa369","sha256:64865480c4690355578843a0309b2e65eb6d0f2a6a0e2ae8f7ef984981bd3f03"],"state_sha256":"c5687d05eea856f62a8faa1807a6154512546c9ea8f27717cd4bafebc6edf380"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vS9tDeCHaBK5TStCBcKwHV8gQ1+XaS3VyaW9z9nHNoeeyytSPUxTPGExIkIbeqzrnb3R3e+f34ge2SU+YYJ9Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T21:49:05.794108Z","bundle_sha256":"1df518e1fe5f6754caccd83c1957749716e5a27492ac7d1414002377b5ea3b51"}}