{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:M7KSSKHPJQH2OANSEOIEIKDN6X","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":"adb6949066e7833762ee502f8eb7f2c6d6ff6d2f354f73d5bb18075cd1ff57c3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-06-29T13:04:23Z","title_canon_sha256":"ab5ae089b0f42472acf391017e1ff72076643c43fc484e928725d1a056ff19d2"},"schema_version":"1.0","source":{"id":"1807.00690","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.00690","created_at":"2026-05-18T00:11:51Z"},{"alias_kind":"arxiv_version","alias_value":"1807.00690v1","created_at":"2026-05-18T00:11:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.00690","created_at":"2026-05-18T00:11:51Z"},{"alias_kind":"pith_short_12","alias_value":"M7KSSKHPJQH2","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"M7KSSKHPJQH2OANS","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"M7KSSKHP","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:3376b0841a7eb16c979a1a614e1bdabaef6f32ec3a54f2259639498c17c6ff23","target":"graph","created_at":"2026-05-18T00:11:51Z","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 $\\alpha\\geq 2$ be any ordinal. We consider the class $\\mathsf{Drs}_{\\alpha}$ of relativized diagonal free set algebras of dimension $\\alpha$. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of $\\mathsf{Drs}_{\\alpha}$ are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class $\\mathsf{Drs}_{\\alpha}$ corresponds to first order logic, without equality symbol, with $\\alpha$-many variables and on relativized semantics. Hence, in this variati","authors_text":"Amitayu Banerjee, Mohamed Khaled","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-06-29T13:04:23Z","title":"First order logic without equality on relativized semantics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.00690","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:ea079ab684d8b3bc427afe4219022c6fcea3228ca6b13da2459fec5aa5e71bb7","target":"record","created_at":"2026-05-18T00:11:51Z","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":"adb6949066e7833762ee502f8eb7f2c6d6ff6d2f354f73d5bb18075cd1ff57c3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-06-29T13:04:23Z","title_canon_sha256":"ab5ae089b0f42472acf391017e1ff72076643c43fc484e928725d1a056ff19d2"},"schema_version":"1.0","source":{"id":"1807.00690","kind":"arxiv","version":1}},"canonical_sha256":"67d52928ef4c0fa701b2239044286df5ee8e0ecc3c9121f703936fe15dc36554","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"67d52928ef4c0fa701b2239044286df5ee8e0ecc3c9121f703936fe15dc36554","first_computed_at":"2026-05-18T00:11:51.771722Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:11:51.771722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9MvECLRVjXAVZy1luxLFW0cBjwyDbA3lPZoK0WAaQ2ca/vQvpEH42LwDoItScUNmrY/QUxXhzuToBE0Q4phbDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:11:51.772557Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.00690","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea079ab684d8b3bc427afe4219022c6fcea3228ca6b13da2459fec5aa5e71bb7","sha256:3376b0841a7eb16c979a1a614e1bdabaef6f32ec3a54f2259639498c17c6ff23"],"state_sha256":"25fbfa5598dfa03cd9d164b28e95b84d9ca0832a2e34da467b275cb3f2604bf3"}