{"paper":{"title":"On neat atom structures for cylindric like algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Tarek Sayed Ahmed","submitted_at":"2013-04-23T12:17:12Z","abstract_excerpt":"(1) Let 1\\leq k\\leq \\omega. Call an atom structure \\alpha weakly k neat representable, the term algebra is in \\RCA_n\\cap \\Nr_n\\CA_{n+k}, but the complex algebra is not representable. Call an atom structure neat if there is an atomic algebra \\A, such that \\At\\A=\\alpha, \\A\\in \\Nr_n\\CA_{\\omega} and for every algebra $\\B$ based on this atom structure there exists k\\in \\omega$, k\\geq 1, such that \\B\\in \\Nr_n\\CA_{n+k}.\n  (2) Let k\\leq \\omega. Call an atom structure \\alpha k complete, if there exists \\A such that \\At\\A=\\alpha and \\A\\in S_c\\Nr_n\\CA_{n+k}.\n  (3) Let k\\leq \\omega. Call an atom structure"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.5151","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"}