Complete Representations and Neat Embeddings - Tarek Sayed Ahmed (Bulletin of the Section of Logic, 2022)
Let \(2<n<ω\). Then \({\sf CA}_n\) denotes the class of cylindric algebras of dimension \(n\), \({\sf RCA}_n\) denotes the class of representable \(\sf CA_n\)s, \({\sf CRCA}_n\) denotes the class of completely representable \({\sf CA}_n\)s, and \({\sf Nr}_n{\sf CA}_{ω}(\subseteq {\sf CA}_n\)) denotes the class of \(n\)-neat reducts of \({\sf CA}_{ω}\)s. The elementary closure of the class \({\sf CRCA}_n\)s (\(K_n\)) and the non-elementary class \({\sf At}({\sf Nr}_n{\sf CA}_{ω})\) are characterized using two-player zero-sum games, where \({\sf At}\) 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 \(S_c{\sf Nr}_n{\sf CA}_{ω}\) where \(S_c\) is the operation of forming complete subalgebras. For any class \(L\) such that \({\sf At}{\sf Nr}_n{\sf CA}_{ω}\subseteq L\subseteq {\sf At}K_n\), it is proved that \({\bf SP}\mathfrak{Cm}L={\sf RCA}_n\), where \({\sf Cm}\) is the dual operator to \(\sf At\); that of forming complex algebras. It is also shown that any class \(K\) between \({\sf CRCA}_n\cap S_d{\sf Nr}_n{\sf CA}_{ω}\) and \(S_c{\sf Nr}_n{\sf CA}_{n+3}\) 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 \({\sf At}({\sf Nr}_n{\sf CA}_{m})\cap {\sf CRCA}_n\subseteq K\subseteq {\sf At}S_c{\sf Nr}_n{\sf CA}_{l}\), \(K\) is not not first order definable either.