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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"},"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":"1304.2930","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2013-04-09T04:35:03Z","cross_cats_sorted":[],"title_canon_sha256":"8bd2fbb580c8f0e126e85b1c23c38767fbbd1f3075591e5c6bb7a4701e892f22","abstract_canon_sha256":"472ed935181fe68840f37344effcf4cd05064793e3ad252511cea0bd916d6b98"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:27.933323Z","signature_b64":"fXa7WiinKWqVVhxi3vHynsB36DWCSxWfOvNgs4U6DSuZ9d9PRDoHZDtszs9VTYuXgJIXttA7TeeSS85XAvoRAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b23a4fb8132f0f67d5234a494a8586e57dade14f263d10bece9cb981419cbe7b","last_reissued_at":"2026-05-18T03:28:27.932792Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:27.932792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"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}","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Tarek Sayed Ahmed","submitted_at":"2013-04-09T04:35:03Z","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. 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