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He conjectured that, when $\\alpha$ is infinite, there are no more atoms. This conjecture has not been confirmed or denied yet. In this article, we show that Tarski's conjecture is true if $\\mathrm{CA}_{\\alpha}$ is replaced by $\\mathrm{D}_{\\alpha}$, $\\mathrm{G}_{\\alpha}$, but the $m$-generated free $\\mathrm{Crs}_{\\alpha}$ algebra is atomless."},"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":"1903.01527","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-03-04T20:29:51Z","cross_cats_sorted":[],"title_canon_sha256":"4c1104981bf66b60ad1f17923c72e156a7eb4e4c4b18c67ae6d0d0d83aabfe9c","abstract_canon_sha256":"f8dee2f274dc593004a3f05085fe1051661105df54654a502b9f488720db6fb1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:00.394290Z","signature_b64":"NjKwMaztb6xqj9gs47leHX1PAbtQmovae998Z47/3+5rNTrJKx4NsJJJ8yv8f3vn+TauZWDeWQjgKsbCxs5eAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3016287de6340fdeabd804b2ce576cc1c094ca3652cd8769afef0c2962877f9f","last_reissued_at":"2026-05-17T23:52:00.393857Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:00.393857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Atoms in infinite dimensional free sequence-set algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Istv\\'an N\\'emeti, Mohamed Khaled","submitted_at":"2019-03-04T20:29:51Z","abstract_excerpt":"A. Tarski proved that the m-generated free algebra of $\\mathrm{CA}_{\\alpha}$, the class of cylindric algebras of dimension $\\alpha$, contains exactly $2^m$ zero-dimensional atoms, when $m\\ge 1$ is a finite cardinal and $\\alpha$ is an arbitrary ordinal. He conjectured that, when $\\alpha$ is infinite, there are no more atoms. This conjecture has not been confirmed or denied yet. In this article, we show that Tarski's conjecture is true if $\\mathrm{CA}_{\\alpha}$ is replaced by $\\mathrm{D}_{\\alpha}$, $\\mathrm{G}_{\\alpha}$, but the $m$-generated free $\\mathrm{Crs}_{\\alpha}$ algebra is atomless."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.01527","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":"1903.01527","created_at":"2026-05-17T23:52:00.393926+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.01527v1","created_at":"2026-05-17T23:52:00.393926+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.01527","created_at":"2026-05-17T23:52:00.393926+00:00"},{"alias_kind":"pith_short_12","alias_value":"GALCQ7PGGQH5","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_16","alias_value":"GALCQ7PGGQH55K6Y","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_8","alias_value":"GALCQ7PG","created_at":"2026-05-18T12:33:18.533446+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/GALCQ7PGGQH55K6YASZM4V3MYH","json":"https://pith.science/pith/GALCQ7PGGQH55K6YASZM4V3MYH.json","graph_json":"https://pith.science/api/pith-number/GALCQ7PGGQH55K6YASZM4V3MYH/graph.json","events_json":"https://pith.science/api/pith-number/GALCQ7PGGQH55K6YASZM4V3MYH/events.json","paper":"https://pith.science/paper/GALCQ7PG"},"agent_actions":{"view_html":"https://pith.science/pith/GALCQ7PGGQH55K6YASZM4V3MYH","download_json":"https://pith.science/pith/GALCQ7PGGQH55K6YASZM4V3MYH.json","view_paper":"https://pith.science/paper/GALCQ7PG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.01527&json=true","fetch_graph":"https://pith.science/api/pith-number/GALCQ7PGGQH55K6YASZM4V3MYH/graph.json","fetch_events":"https://pith.science/api/pith-number/GALCQ7PGGQH55K6YASZM4V3MYH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GALCQ7PGGQH55K6YASZM4V3MYH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GALCQ7PGGQH55K6YASZM4V3MYH/action/storage_attestation","attest_author":"https://pith.science/pith/GALCQ7PGGQH55K6YASZM4V3MYH/action/author_attestation","sign_citation":"https://pith.science/pith/GALCQ7PGGQH55K6YASZM4V3MYH/action/citation_signature","submit_replication":"https://pith.science/pith/GALCQ7PGGQH55K6YASZM4V3MYH/action/replication_record"}},"created_at":"2026-05-17T23:52:00.393926+00:00","updated_at":"2026-05-17T23:52:00.393926+00:00"}