{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:TEMX47C665UUOLDI27A74Z2KFT","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":"f6f06e04b42f7842e026b5f29fd45bb0987a77b6ee0b89b0b4c31d006ab1fc48","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-15T08:13:25Z","title_canon_sha256":"ebd97b8213ff40c5442c592a0d92e0a211c14852631316698ba8169071a4b1d0"},"schema_version":"1.0","source":{"id":"1705.05781","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.05781","created_at":"2026-05-18T00:44:20Z"},{"alias_kind":"arxiv_version","alias_value":"1705.05781v1","created_at":"2026-05-18T00:44:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.05781","created_at":"2026-05-18T00:44:20Z"},{"alias_kind":"pith_short_12","alias_value":"TEMX47C665UU","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_16","alias_value":"TEMX47C665UUOLDI","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_8","alias_value":"TEMX47C6","created_at":"2026-05-18T12:31:46Z"}],"graph_snapshots":[{"event_id":"sha256:7265fe908dbe162472b3e8e0acba88565f748b204b0ad1bd4647fa5c41de6a3a","target":"graph","created_at":"2026-05-18T00:44:20Z","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 modular semilattice is a semilattice generalization of a modular lattice. We establish a Birkhoff-type representation theorem for modular semilattices, which says that every modular semilattice is isomorphic to the family of ideals in a certain poset with additional relations.This new poset structure, which we axiomatize in this paper, is called a PPIP (projective poset with inconsistent pairs). A PPIP is a common generalization of a PIP (poset with inconsistent pairs) and a projective ordered space. The former was introduced by Barth\\'elemy and Constantin for establishing Birkhoff-type theo","authors_text":"Hiroshi Hirai, So Nakashima","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-15T08:13:25Z","title":"A Compact Representation for Modular Semilattices and its Applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.05781","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:d6c135c75fbe8e8c698b57439df5f53597c9234e3e84197e2a733944bab46afc","target":"record","created_at":"2026-05-18T00:44:20Z","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":"f6f06e04b42f7842e026b5f29fd45bb0987a77b6ee0b89b0b4c31d006ab1fc48","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-15T08:13:25Z","title_canon_sha256":"ebd97b8213ff40c5442c592a0d92e0a211c14852631316698ba8169071a4b1d0"},"schema_version":"1.0","source":{"id":"1705.05781","kind":"arxiv","version":1}},"canonical_sha256":"99197e7c5ef769472c68d7c1fe674a2ce2a4c8cb5c23e74e81cd9f84b752cd0b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99197e7c5ef769472c68d7c1fe674a2ce2a4c8cb5c23e74e81cd9f84b752cd0b","first_computed_at":"2026-05-18T00:44:20.185756Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:20.185756Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KNN0NNWk7z6TtcBS35VSlrd3AMtb/hq8swKhIk44P30X7qa49zfIlBtDazDTvRQaTyqawSFsY3F1EMhA2MjpAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:20.186261Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.05781","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d6c135c75fbe8e8c698b57439df5f53597c9234e3e84197e2a733944bab46afc","sha256:7265fe908dbe162472b3e8e0acba88565f748b204b0ad1bd4647fa5c41de6a3a"],"state_sha256":"a35a8295b5464212dccdd9d017ebf9bc6786199744183c932b8234f695119c2d"}