{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:WI5E7OATF4HWPVJDJJEUVBMG4V","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":"472ed935181fe68840f37344effcf4cd05064793e3ad252511cea0bd916d6b98","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2013-04-09T04:35:03Z","title_canon_sha256":"8bd2fbb580c8f0e126e85b1c23c38767fbbd1f3075591e5c6bb7a4701e892f22"},"schema_version":"1.0","source":{"id":"1304.2930","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.2930","created_at":"2026-05-18T03:28:27Z"},{"alias_kind":"arxiv_version","alias_value":"1304.2930v1","created_at":"2026-05-18T03:28:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.2930","created_at":"2026-05-18T03:28:27Z"},{"alias_kind":"pith_short_12","alias_value":"WI5E7OATF4HW","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"WI5E7OATF4HWPVJD","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"WI5E7OAT","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:78e461c1ff179eaba1751b86742d6b3e84fc8bcdb750745e7288451a7bac66c6","target":"graph","created_at":"2026-05-18T03:28:27Z","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":"We show that for 1<n<m, the class Nr_nCA_m known to be non-elementary is pseudo elementary. When n and m are finite we use a two sorted theory, when n is finite and m infinite we use a three sorted one, and finally when both are infinite we use a four sorted defining theory. Our non finite axiomatizability result, follows from the fact that for 2<n<m, and any r\\in \\omega there exists a finite (Monk like) algebra C(m,n,r), such that C(m,n,r)\\in Nr_nCA_m C(m,n,r)\\notin SNr_nCA_{m+1}, and any non trivial ultraproduct on r of such algebras in in ElNr_nCA_m. Finally we use such algebras, to show th","authors_text":"Tarek Sayed Ahmed","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2013-04-09T04:35:03Z","title":"The elementary closure of the class Nr_nCA_m for m\\geq n+1 is not finitely axiomatizable, futhermore for any finite k\\geq 1, there is A\\in Nr_{\\omega}CA_{\\omeg+k}that is not SNr_{\\omega}CA_{\\omega+k+1}"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.2930","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":"sha256:3e27dace5c62d258eef4783610334564e3edf2e9c899aca6386161ecf7d65f39","target":"record","created_at":"2026-05-18T03:28:27Z","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":"472ed935181fe68840f37344effcf4cd05064793e3ad252511cea0bd916d6b98","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2013-04-09T04:35:03Z","title_canon_sha256":"8bd2fbb580c8f0e126e85b1c23c38767fbbd1f3075591e5c6bb7a4701e892f22"},"schema_version":"1.0","source":{"id":"1304.2930","kind":"arxiv","version":1}},"canonical_sha256":"b23a4fb8132f0f67d5234a494a8586e57dade14f263d10bece9cb981419cbe7b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b23a4fb8132f0f67d5234a494a8586e57dade14f263d10bece9cb981419cbe7b","first_computed_at":"2026-05-18T03:28:27.932792Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:28:27.932792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fXa7WiinKWqVVhxi3vHynsB36DWCSxWfOvNgs4U6DSuZ9d9PRDoHZDtszs9VTYuXgJIXttA7TeeSS85XAvoRAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:28:27.933323Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.2930","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e27dace5c62d258eef4783610334564e3edf2e9c899aca6386161ecf7d65f39","sha256:78e461c1ff179eaba1751b86742d6b3e84fc8bcdb750745e7288451a7bac66c6"],"state_sha256":"e4e1a98ec368da5f4c29b16e4d2fe40965572429174b6190561b064715b4a4fb"}