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When we consider dcpos (directed complete posets) equipped their Scott topologies, a similar question arises: which dcpos $P$ have the property that for any dcpo $Q$, $C_\\sigma(P)$ isomorphic to $C_\\sigma(Q)$ implies $P$ is isomorphic to $Q$ (such a dcpo $P$ will be called Scott closed set lattice faithful, or SCL-faithful in short)? Here $C_{\\sigma}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1607.03576","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-07-13T02:45:45Z","cross_cats_sorted":[],"title_canon_sha256":"cd084d71e5db0892bc1aa1023a186d4cdb65ae185e6540f58af16056a923f8c0","abstract_canon_sha256":"84eef59a76df2448bd7e9f04d2738c874e58df39fd110b59d73a75df7efce93e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:07.651882Z","signature_b64":"XXaO8dsKsPmz5A5qga/2wBL6PfnQtwSCY1kR8i4m3FiQp1tatxVLRO/BkLU0EUw98lIZd9bxiU9CkIgn5/WCCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e005154097c18c6db25523dbe1978451e4b010af0ee619ca913a8384ea1b64b","last_reissued_at":"2026-05-18T01:11:07.651442Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:07.651442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Faithfulness of Directed Complete Posets based on Scott Closed Set Lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Dongsheng Zhao, Luoshan Xu","submitted_at":"2016-07-13T02:45:45Z","abstract_excerpt":"By Thron, a topological space $X$ has the property that $C(X)$ isomorphic to $C(Y)$ implies $X$ is homeomorphic to $Y$ iff $X$ is sober and $T_D$, where $C(X)$ and $C(Y)$ denote the lattices of closed sets of $X$ and $T_0$ space $Y$, respectively. When we consider dcpos (directed complete posets) equipped their Scott topologies, a similar question arises: which dcpos $P$ have the property that for any dcpo $Q$, $C_\\sigma(P)$ isomorphic to $C_\\sigma(Q)$ implies $P$ is isomorphic to $Q$ (such a dcpo $P$ will be called Scott closed set lattice faithful, or SCL-faithful in short)? Here $C_{\\sigma}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.03576","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1607.03576","created_at":"2026-05-18T01:11:07.651512+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.03576v1","created_at":"2026-05-18T01:11:07.651512+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.03576","created_at":"2026-05-18T01:11:07.651512+00:00"},{"alias_kind":"pith_short_12","alias_value":"PYAFCVAJPQMM","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_16","alias_value":"PYAFCVAJPQMMNWZF","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_8","alias_value":"PYAFCVAJ","created_at":"2026-05-18T12:30:39.010887+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PYAFCVAJPQMMNWZFKI634GLYIU","json":"https://pith.science/pith/PYAFCVAJPQMMNWZFKI634GLYIU.json","graph_json":"https://pith.science/api/pith-number/PYAFCVAJPQMMNWZFKI634GLYIU/graph.json","events_json":"https://pith.science/api/pith-number/PYAFCVAJPQMMNWZFKI634GLYIU/events.json","paper":"https://pith.science/paper/PYAFCVAJ"},"agent_actions":{"view_html":"https://pith.science/pith/PYAFCVAJPQMMNWZFKI634GLYIU","download_json":"https://pith.science/pith/PYAFCVAJPQMMNWZFKI634GLYIU.json","view_paper":"https://pith.science/paper/PYAFCVAJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.03576&json=true","fetch_graph":"https://pith.science/api/pith-number/PYAFCVAJPQMMNWZFKI634GLYIU/graph.json","fetch_events":"https://pith.science/api/pith-number/PYAFCVAJPQMMNWZFKI634GLYIU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PYAFCVAJPQMMNWZFKI634GLYIU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PYAFCVAJPQMMNWZFKI634GLYIU/action/storage_attestation","attest_author":"https://pith.science/pith/PYAFCVAJPQMMNWZFKI634GLYIU/action/author_attestation","sign_citation":"https://pith.science/pith/PYAFCVAJPQMMNWZFKI634GLYIU/action/citation_signature","submit_replication":"https://pith.science/pith/PYAFCVAJPQMMNWZFKI634GLYIU/action/replication_record"}},"created_at":"2026-05-18T01:11:07.651512+00:00","updated_at":"2026-05-18T01:11:07.651512+00:00"}