{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:QVKNDDP5BOSAX2HJ3LHN7AUH7R","short_pith_number":"pith:QVKNDDP5","canonical_record":{"source":{"id":"1203.0765","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-03-04T19:30:20Z","cross_cats_sorted":[],"title_canon_sha256":"d6184a7f8102e8a9e958e932f13c1a5506b5203fafc11080fe93da82f18917d6","abstract_canon_sha256":"c644d25f09db7781f501a6bd1c9e6066bf860f147b79a0a9d2fc58903cf69378"},"schema_version":"1.0"},"canonical_sha256":"8554d18dfd0ba40be8e9dacedf8287fc48256d788de5ea717b068ae10a51973e","source":{"kind":"arxiv","id":"1203.0765","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.0765","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"arxiv_version","alias_value":"1203.0765v1","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.0765","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"pith_short_12","alias_value":"QVKNDDP5BOSA","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"QVKNDDP5BOSAX2HJ","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"QVKNDDP5","created_at":"2026-05-18T12:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:QVKNDDP5BOSAX2HJ3LHN7AUH7R","target":"record","payload":{"canonical_record":{"source":{"id":"1203.0765","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-03-04T19:30:20Z","cross_cats_sorted":[],"title_canon_sha256":"d6184a7f8102e8a9e958e932f13c1a5506b5203fafc11080fe93da82f18917d6","abstract_canon_sha256":"c644d25f09db7781f501a6bd1c9e6066bf860f147b79a0a9d2fc58903cf69378"},"schema_version":"1.0"},"canonical_sha256":"8554d18dfd0ba40be8e9dacedf8287fc48256d788de5ea717b068ae10a51973e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:41.014525Z","signature_b64":"itVroKSPz13yeGnOoSGUQ8e9G4lMZkrt8AlF7HqxgK3h3/1H/ffoCpmRSM/Vw9RjH1E++QsQwz8iHKDP316MDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8554d18dfd0ba40be8e9dacedf8287fc48256d788de5ea717b068ae10a51973e","last_reissued_at":"2026-05-18T00:53:41.014054Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:41.014054Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.0765","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":"2026-05-18T00:53:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MSKv4etzOt4Z0GrjiT2pQa2oMNjEAe7H/mZNVqhNujRdy+eG5w6rmo0ZZWusc5GM/Q6aeWdBFzaqpEZxVTyhCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T14:04:53.916748Z"},"content_sha256":"4766fe99ad4330bc6574d404f425ccf9da6a0a6e050e9fc9c72cea5a1f22a095","schema_version":"1.0","event_id":"sha256:4766fe99ad4330bc6574d404f425ccf9da6a0a6e050e9fc9c72cea5a1f22a095"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:QVKNDDP5BOSAX2HJ3LHN7AUH7R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Atomistic subsemirings of the lattice of subspaces of an algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Daniel S. Sage","submitted_at":"2012-03-04T19:30:20Z","abstract_excerpt":"Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1_A is the set of atoms of a subalgebra of A called the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.0765","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":"2026-05-18T00:53:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WWnMYgWbTXKTEL3g/+VkcjoDY75o5lvMh6VJktulLQWOdoYqEVXNFGzRXRYTfa4feFifjlc0P0/Tux+wnJ7gDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T14:04:53.917103Z"},"content_sha256":"39d5037bca1129c8ce010a031265d850e630692f512e0704a4ffd0c38b1ae39d","schema_version":"1.0","event_id":"sha256:39d5037bca1129c8ce010a031265d850e630692f512e0704a4ffd0c38b1ae39d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QVKNDDP5BOSAX2HJ3LHN7AUH7R/bundle.json","state_url":"https://pith.science/pith/QVKNDDP5BOSAX2HJ3LHN7AUH7R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QVKNDDP5BOSAX2HJ3LHN7AUH7R/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":"2026-06-05T14:04:53Z","links":{"resolver":"https://pith.science/pith/QVKNDDP5BOSAX2HJ3LHN7AUH7R","bundle":"https://pith.science/pith/QVKNDDP5BOSAX2HJ3LHN7AUH7R/bundle.json","state":"https://pith.science/pith/QVKNDDP5BOSAX2HJ3LHN7AUH7R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QVKNDDP5BOSAX2HJ3LHN7AUH7R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:QVKNDDP5BOSAX2HJ3LHN7AUH7R","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":"c644d25f09db7781f501a6bd1c9e6066bf860f147b79a0a9d2fc58903cf69378","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-03-04T19:30:20Z","title_canon_sha256":"d6184a7f8102e8a9e958e932f13c1a5506b5203fafc11080fe93da82f18917d6"},"schema_version":"1.0","source":{"id":"1203.0765","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.0765","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"arxiv_version","alias_value":"1203.0765v1","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.0765","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"pith_short_12","alias_value":"QVKNDDP5BOSA","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"QVKNDDP5BOSAX2HJ","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"QVKNDDP5","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:39d5037bca1129c8ce010a031265d850e630692f512e0704a4ffd0c38b1ae39d","target":"graph","created_at":"2026-05-18T00:53:41Z","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 A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1_A is the set of atoms of a subalgebra of A called the ","authors_text":"Daniel S. Sage","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-03-04T19:30:20Z","title":"Atomistic subsemirings of the lattice of subspaces of an algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.0765","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":"sha256:4766fe99ad4330bc6574d404f425ccf9da6a0a6e050e9fc9c72cea5a1f22a095","target":"record","created_at":"2026-05-18T00:53:41Z","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":"c644d25f09db7781f501a6bd1c9e6066bf860f147b79a0a9d2fc58903cf69378","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-03-04T19:30:20Z","title_canon_sha256":"d6184a7f8102e8a9e958e932f13c1a5506b5203fafc11080fe93da82f18917d6"},"schema_version":"1.0","source":{"id":"1203.0765","kind":"arxiv","version":1}},"canonical_sha256":"8554d18dfd0ba40be8e9dacedf8287fc48256d788de5ea717b068ae10a51973e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8554d18dfd0ba40be8e9dacedf8287fc48256d788de5ea717b068ae10a51973e","first_computed_at":"2026-05-18T00:53:41.014054Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:41.014054Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"itVroKSPz13yeGnOoSGUQ8e9G4lMZkrt8AlF7HqxgK3h3/1H/ffoCpmRSM/Vw9RjH1E++QsQwz8iHKDP316MDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:41.014525Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.0765","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4766fe99ad4330bc6574d404f425ccf9da6a0a6e050e9fc9c72cea5a1f22a095","sha256:39d5037bca1129c8ce010a031265d850e630692f512e0704a4ffd0c38b1ae39d"],"state_sha256":"522add14cf124a4aa898f16500490a32b375e374b0ed6e7f8574901558e3ac1f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qPjby0ZANdzUhoNESypFsQnIJZ8lsrWqF+Ihm0xzfBtU055gPTIZAQhzB8OfnqXcY1K0CAXiSKdaABeaNvHVAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T14:04:53.918985Z","bundle_sha256":"722d9c40b59fb6f90d4bf333f33f848a97ab48242650f7aa48c74266bf450e43"}}