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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"},"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":"1807.00690","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-06-29T13:04:23Z","cross_cats_sorted":[],"title_canon_sha256":"ab5ae089b0f42472acf391017e1ff72076643c43fc484e928725d1a056ff19d2","abstract_canon_sha256":"adb6949066e7833762ee502f8eb7f2c6d6ff6d2f354f73d5bb18075cd1ff57c3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:51.772557Z","signature_b64":"9MvECLRVjXAVZy1luxLFW0cBjwyDbA3lPZoK0WAaQ2ca/vQvpEH42LwDoItScUNmrY/QUxXhzuToBE0Q4phbDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67d52928ef4c0fa701b2239044286df5ee8e0ecc3c9121f703936fe15dc36554","last_reissued_at":"2026-05-18T00:11:51.771722Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:51.771722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"First order logic without equality on relativized semantics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Amitayu Banerjee, Mohamed Khaled","submitted_at":"2018-06-29T13:04:23Z","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. 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