Complete Representations and Neat Embeddings - Sayed Ahmed, Tarek (Bulletin of the Section of Logic, 2022)
Let \(2<n<ω\). Then \(CA_n\) denotes the class of cylindric algebras of dimension \(n\), \(RCA_n\) denotes the class of representable \(CA_n\)s, \(CRCA_n\) denotes the class of completely representable \(CA_n\)s, and \(Nr_nCA_ω(CA_n\)) denotes the class of \(n\)-neat reducts of \(CA_ω\)s. The elementary closure of the class \(CRCA_n\)s (\(K_n\)) and the non-elementary class \(At(Nr_nCA_ω)\) are characterized using two-player zero-sum games, where \(At\) is the operator of forming atom structures. It is shown that \(K_n\) is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of \(S_cNr_nCA_ω\) where \(S_c\) is the operation of forming complete subalgebras. For any class \(L\) such that \(AtNr_nCA_ωLAtK_n\), it is proved that \(SPCmL=RCA_n\), where \(Cm\) is the dual operator to \(At\); that of forming complex algebras. It is also shown that any class \(K\) between \(CRCA_nS_dNr_nCA_ω\) and \(S_cNr_nCA_n+3\) is not first order definable, where \(S_d\) is the operation of forming dense subalgebras, and that for any \(2<n<m\), any \(l≥ n+3\) any any class \(K\) (such that \(At(Nr_nCA_m)CRCA_nKAtS_cNr_nCA_l\), \(K\) is not not first order definable either.