{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:TVZXZ7FXHEMSZRK4UFDLR6KZQG","short_pith_number":"pith:TVZXZ7FX","canonical_record":{"source":{"id":"1102.2195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-02-08T07:25:11Z","cross_cats_sorted":[],"title_canon_sha256":"29eb75ef8d4654a6a4749eca3675269dc3873461044659b034fc4660058d0013","abstract_canon_sha256":"4f3cdaea8f21925edb90931883363d4deaf8483f020264cde232930278af411b"},"schema_version":"1.0"},"canonical_sha256":"9d737cfcb739192cc55ca146b8f95981973374bf77ecf00b131da0ca0d83a5f1","source":{"kind":"arxiv","id":"1102.2195","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.2195","created_at":"2026-05-18T04:19:02Z"},{"alias_kind":"arxiv_version","alias_value":"1102.2195v2","created_at":"2026-05-18T04:19:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.2195","created_at":"2026-05-18T04:19:02Z"},{"alias_kind":"pith_short_12","alias_value":"TVZXZ7FXHEMS","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TVZXZ7FXHEMSZRK4","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TVZXZ7FX","created_at":"2026-05-18T12:26:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:TVZXZ7FXHEMSZRK4UFDLR6KZQG","target":"record","payload":{"canonical_record":{"source":{"id":"1102.2195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-02-08T07:25:11Z","cross_cats_sorted":[],"title_canon_sha256":"29eb75ef8d4654a6a4749eca3675269dc3873461044659b034fc4660058d0013","abstract_canon_sha256":"4f3cdaea8f21925edb90931883363d4deaf8483f020264cde232930278af411b"},"schema_version":"1.0"},"canonical_sha256":"9d737cfcb739192cc55ca146b8f95981973374bf77ecf00b131da0ca0d83a5f1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:02.144025Z","signature_b64":"o8ge2MIhaTwGbLB0B4DKqfch8EJNADLOIAXFP6IDhF8tSbF7rW7gZIRCz9aSUvEqDKfUBtzeOh+PWzKvraXeDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d737cfcb739192cc55ca146b8f95981973374bf77ecf00b131da0ca0d83a5f1","last_reissued_at":"2026-05-18T04:19:02.143445Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:02.143445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.2195","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":"2026-05-18T04:19:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g/SAFcCz/N9iQljnYpCEgRGjrI4YbTkw30YFosoLQ3yU5MWzIcqNo5S0SnPhfO0euq0yAaYfELuU1l4AEQzqDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T13:15:57.118658Z"},"content_sha256":"35289fa79cf7b1d7aea49b082c813d3236243b595032d8274201725de381c864","schema_version":"1.0","event_id":"sha256:35289fa79cf7b1d7aea49b082c813d3236243b595032d8274201725de381c864"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:TVZXZ7FXHEMSZRK4UFDLR6KZQG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Varieties of lattices with geometric descriptions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Friedrich Wehrung (LMNO), Luigi Santocanale (LIF)","submitted_at":"2011-02-08T07:25:11Z","abstract_excerpt":"A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann, Pickering, and Roddy proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem. This solves two problems"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.2195","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":"2026-05-18T04:19:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ehny4W1QtUzwDpov9BWRKPCIEnqwFU/BzwKv51h1YAoKbHotMv+QwB4q5pYT6iBYVKfcHG9T1rYBiwZ/F7UhBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T13:15:57.119011Z"},"content_sha256":"d7b3782abece358853e67fc354115076e798e118e77334479575a33097e53840","schema_version":"1.0","event_id":"sha256:d7b3782abece358853e67fc354115076e798e118e77334479575a33097e53840"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TVZXZ7FXHEMSZRK4UFDLR6KZQG/bundle.json","state_url":"https://pith.science/pith/TVZXZ7FXHEMSZRK4UFDLR6KZQG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TVZXZ7FXHEMSZRK4UFDLR6KZQG/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":"2026-06-09T13:15:57Z","links":{"resolver":"https://pith.science/pith/TVZXZ7FXHEMSZRK4UFDLR6KZQG","bundle":"https://pith.science/pith/TVZXZ7FXHEMSZRK4UFDLR6KZQG/bundle.json","state":"https://pith.science/pith/TVZXZ7FXHEMSZRK4UFDLR6KZQG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TVZXZ7FXHEMSZRK4UFDLR6KZQG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:TVZXZ7FXHEMSZRK4UFDLR6KZQG","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":"4f3cdaea8f21925edb90931883363d4deaf8483f020264cde232930278af411b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-02-08T07:25:11Z","title_canon_sha256":"29eb75ef8d4654a6a4749eca3675269dc3873461044659b034fc4660058d0013"},"schema_version":"1.0","source":{"id":"1102.2195","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.2195","created_at":"2026-05-18T04:19:02Z"},{"alias_kind":"arxiv_version","alias_value":"1102.2195v2","created_at":"2026-05-18T04:19:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.2195","created_at":"2026-05-18T04:19:02Z"},{"alias_kind":"pith_short_12","alias_value":"TVZXZ7FXHEMS","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TVZXZ7FXHEMSZRK4","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TVZXZ7FX","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:d7b3782abece358853e67fc354115076e798e118e77334479575a33097e53840","target":"graph","created_at":"2026-05-18T04:19:02Z","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":"A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann, Pickering, and Roddy proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem. This solves two problems","authors_text":"Friedrich Wehrung (LMNO), Luigi Santocanale (LIF)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-02-08T07:25:11Z","title":"Varieties of lattices with geometric descriptions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.2195","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":"sha256:35289fa79cf7b1d7aea49b082c813d3236243b595032d8274201725de381c864","target":"record","created_at":"2026-05-18T04:19:02Z","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":"4f3cdaea8f21925edb90931883363d4deaf8483f020264cde232930278af411b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-02-08T07:25:11Z","title_canon_sha256":"29eb75ef8d4654a6a4749eca3675269dc3873461044659b034fc4660058d0013"},"schema_version":"1.0","source":{"id":"1102.2195","kind":"arxiv","version":2}},"canonical_sha256":"9d737cfcb739192cc55ca146b8f95981973374bf77ecf00b131da0ca0d83a5f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d737cfcb739192cc55ca146b8f95981973374bf77ecf00b131da0ca0d83a5f1","first_computed_at":"2026-05-18T04:19:02.143445Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:19:02.143445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o8ge2MIhaTwGbLB0B4DKqfch8EJNADLOIAXFP6IDhF8tSbF7rW7gZIRCz9aSUvEqDKfUBtzeOh+PWzKvraXeDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:19:02.144025Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.2195","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:35289fa79cf7b1d7aea49b082c813d3236243b595032d8274201725de381c864","sha256:d7b3782abece358853e67fc354115076e798e118e77334479575a33097e53840"],"state_sha256":"6dab38e33bdd381fd4ddc1ba99800ab5d011d307a363634bee8866c153714169"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y8SRynNKtip46Fo203QXw0Fe8lPec8iSm62pUGD8uvti07HPmB1D3bBINZ60Ial6NuuvxUFqK1lTftLaRbrJDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T13:15:57.121588Z","bundle_sha256":"55a01ecf93372bef9efff5fc68e27911ad3f05f288f3fb864657feedc6af6a60"}}