{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:XANTTSDD5IFIFDYAQYZ25OGT5K","short_pith_number":"pith:XANTTSDD","canonical_record":{"source":{"id":"1807.06862","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-07-18T11:06:14Z","cross_cats_sorted":["cs.LO","math.CO"],"title_canon_sha256":"b6505acaa617effd9a0a7e4be89b4a44c53a34e472a10087c7e17f981ccc0cd9","abstract_canon_sha256":"3e2c1cd5ed803af3fb0b2ba4a8a0fdeaf1230ff2775f71a4a9b278ce5a4b6f53"},"schema_version":"1.0"},"canonical_sha256":"b81b39c863ea0a828f008633aeb8d3eaa47a2586a5eb31e94ebf43b0031f774a","source":{"kind":"arxiv","id":"1807.06862","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.06862","created_at":"2026-05-18T00:10:25Z"},{"alias_kind":"arxiv_version","alias_value":"1807.06862v1","created_at":"2026-05-18T00:10:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.06862","created_at":"2026-05-18T00:10:25Z"},{"alias_kind":"pith_short_12","alias_value":"XANTTSDD5IFI","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"XANTTSDD5IFIFDYA","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"XANTTSDD","created_at":"2026-05-18T12:33:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:XANTTSDD5IFIFDYAQYZ25OGT5K","target":"record","payload":{"canonical_record":{"source":{"id":"1807.06862","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-07-18T11:06:14Z","cross_cats_sorted":["cs.LO","math.CO"],"title_canon_sha256":"b6505acaa617effd9a0a7e4be89b4a44c53a34e472a10087c7e17f981ccc0cd9","abstract_canon_sha256":"3e2c1cd5ed803af3fb0b2ba4a8a0fdeaf1230ff2775f71a4a9b278ce5a4b6f53"},"schema_version":"1.0"},"canonical_sha256":"b81b39c863ea0a828f008633aeb8d3eaa47a2586a5eb31e94ebf43b0031f774a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:25.522864Z","signature_b64":"qCtg7ixYNi5/9EwHqf1a9UbNaP3GfOpYOqu41kBs9Y7kw8n4LORFpXb4KWDReBhQR+1Ra/hU+/C4QedUj2g6Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b81b39c863ea0a828f008633aeb8d3eaa47a2586a5eb31e94ebf43b0031f774a","last_reissued_at":"2026-05-18T00:10:25.522225Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:25.522225Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.06862","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":"2026-05-18T00:10:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ui2ndOWEzCsExbSwqb8+nyjTe+qeHQKDevbw67q51KLbUVQYt/bCUOL6N6jXkT+pCMRILwplCGyrmJPQ9FHRDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T22:28:02.943419Z"},"content_sha256":"9b935e65b2ce8a05c0fa14c3cb7fcf24bb77bd5528035816330aaf9440d0cba8","schema_version":"1.0","event_id":"sha256:9b935e65b2ce8a05c0fa14c3cb7fcf24bb77bd5528035816330aaf9440d0cba8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:XANTTSDD5IFIFDYAQYZ25OGT5K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mix $\\star$-autonomous quantales and the continuous weak order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LO","math.CO"],"primary_cat":"math.LO","authors_text":"Luigi Santocanale (LIS), Maria Jo\\~ao Gouveia (ULISBOA)","submitted_at":"2018-07-18T11:06:14Z","abstract_excerpt":"The set of permutations on a finite set can be given a lattice structure (known as the weak Bruhat order). The lattice structure is generalized to the set of words on a fixed alphabet $\\Sigma = \\{ x, y, z, ... \\}$, where each letter has a fixed number of occurrences (these lattices are known as multinomial lattices and, in dimension 2, as lattices of lattice paths). By interpreting the letters $x, y, z, ...$ as axes, these words can be interpreted as discrete increasing paths on a grid of a $d$-dimensional cube, where $d = {\\rm card}(\\Sigma)$. We show in this paper how to extend this order to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.06862","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":"2026-05-18T00:10:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HQnDo/VwU80AlbWi/MxaBH2NRr3Dv29wR+ypwVRh/a7iSQ0tcw96p4aL2uv3P5cDiyLTwIr9WeutvRzx0icWCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T22:28:02.943777Z"},"content_sha256":"2bfebae2d07d733af956a36f565022ede2b8b1c85017ef00d4c6b88701619130","schema_version":"1.0","event_id":"sha256:2bfebae2d07d733af956a36f565022ede2b8b1c85017ef00d4c6b88701619130"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XANTTSDD5IFIFDYAQYZ25OGT5K/bundle.json","state_url":"https://pith.science/pith/XANTTSDD5IFIFDYAQYZ25OGT5K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XANTTSDD5IFIFDYAQYZ25OGT5K/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":"2026-06-21T22:28:02Z","links":{"resolver":"https://pith.science/pith/XANTTSDD5IFIFDYAQYZ25OGT5K","bundle":"https://pith.science/pith/XANTTSDD5IFIFDYAQYZ25OGT5K/bundle.json","state":"https://pith.science/pith/XANTTSDD5IFIFDYAQYZ25OGT5K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XANTTSDD5IFIFDYAQYZ25OGT5K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:XANTTSDD5IFIFDYAQYZ25OGT5K","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":"3e2c1cd5ed803af3fb0b2ba4a8a0fdeaf1230ff2775f71a4a9b278ce5a4b6f53","cross_cats_sorted":["cs.LO","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-07-18T11:06:14Z","title_canon_sha256":"b6505acaa617effd9a0a7e4be89b4a44c53a34e472a10087c7e17f981ccc0cd9"},"schema_version":"1.0","source":{"id":"1807.06862","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.06862","created_at":"2026-05-18T00:10:25Z"},{"alias_kind":"arxiv_version","alias_value":"1807.06862v1","created_at":"2026-05-18T00:10:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.06862","created_at":"2026-05-18T00:10:25Z"},{"alias_kind":"pith_short_12","alias_value":"XANTTSDD5IFI","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"XANTTSDD5IFIFDYA","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"XANTTSDD","created_at":"2026-05-18T12:33:01Z"}],"graph_snapshots":[{"event_id":"sha256:2bfebae2d07d733af956a36f565022ede2b8b1c85017ef00d4c6b88701619130","target":"graph","created_at":"2026-05-18T00:10:25Z","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 set of permutations on a finite set can be given a lattice structure (known as the weak Bruhat order). The lattice structure is generalized to the set of words on a fixed alphabet $\\Sigma = \\{ x, y, z, ... \\}$, where each letter has a fixed number of occurrences (these lattices are known as multinomial lattices and, in dimension 2, as lattices of lattice paths). By interpreting the letters $x, y, z, ...$ as axes, these words can be interpreted as discrete increasing paths on a grid of a $d$-dimensional cube, where $d = {\\rm card}(\\Sigma)$. We show in this paper how to extend this order to ","authors_text":"Luigi Santocanale (LIS), Maria Jo\\~ao Gouveia (ULISBOA)","cross_cats":["cs.LO","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-07-18T11:06:14Z","title":"Mix $\\star$-autonomous quantales and the continuous weak order"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.06862","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":"sha256:9b935e65b2ce8a05c0fa14c3cb7fcf24bb77bd5528035816330aaf9440d0cba8","target":"record","created_at":"2026-05-18T00:10:25Z","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":"3e2c1cd5ed803af3fb0b2ba4a8a0fdeaf1230ff2775f71a4a9b278ce5a4b6f53","cross_cats_sorted":["cs.LO","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-07-18T11:06:14Z","title_canon_sha256":"b6505acaa617effd9a0a7e4be89b4a44c53a34e472a10087c7e17f981ccc0cd9"},"schema_version":"1.0","source":{"id":"1807.06862","kind":"arxiv","version":1}},"canonical_sha256":"b81b39c863ea0a828f008633aeb8d3eaa47a2586a5eb31e94ebf43b0031f774a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b81b39c863ea0a828f008633aeb8d3eaa47a2586a5eb31e94ebf43b0031f774a","first_computed_at":"2026-05-18T00:10:25.522225Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:25.522225Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qCtg7ixYNi5/9EwHqf1a9UbNaP3GfOpYOqu41kBs9Y7kw8n4LORFpXb4KWDReBhQR+1Ra/hU+/C4QedUj2g6Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:25.522864Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.06862","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9b935e65b2ce8a05c0fa14c3cb7fcf24bb77bd5528035816330aaf9440d0cba8","sha256:2bfebae2d07d733af956a36f565022ede2b8b1c85017ef00d4c6b88701619130"],"state_sha256":"45ee1dab0434d149e5d6a5e06bcc66437a3aa100ca41082be2b8a32611341e12"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K/2PgR01i7amjud/4l+Orl5qVhUgbOuupkm8OkPe6EBfyMwoPGrCK1KMNzSk2BieINCc9/uEgGZNrIomyDpiDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T22:28:02.946119Z","bundle_sha256":"8ae63732a317b2ddde21b0738844cd20d32bfb993e5e35aa1ec47963f62e3aad"}}