Complete Representations and Neat Embeddings

Abstract

Let \(2<n<\omega\). 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 \({\sf Nr}_n{\sf CA}_{\omega}(\subseteq {\sf CA}_n\)) denotes the class of \(n\)-neat reducts of \({\sf CA}_{\omega}\)s. The elementary closure of the class \({\sf CRCA}_n\)s (\(\mathbf{K_n}\)) and the non-elementary class \({\sf At}({\sf Nr}_n{\sf CA}_{\omega})\) are characterized using two-player zero-sum games, where \({\sf At}\) is the operator of forming atom structures. It is shown that \(\mathbf{K_n}\) is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of \(\mathbf{S_c}{\sf Nr}_n{\sf CA}_{\omega}\) where \(\mathbf{S_c}\) is the operation of forming complete subalgebras. For any class \(\mathbf{L}\) such that \({\sf At}{\sf Nr}_n{\sf CA}_{\omega}\subseteq \mathbf{L}\subseteq {\sf At}\mathbf{K_n}\), it is proved that \({\bf SP}\mathfrak{Cm}\mathbf{L}={\sf RCA}_n\), where \({\sf Cm}\) is the dual operator to \(\sf At\); that of forming complex algebras. It is also shown that any class \(\mathbf{K}\) between \({\sf CRCA}_n\cap \mathbf{S_d}{\sf Nr}_n{\sf CA}_{\omega}\) and \(\mathbf{S_c}{\sf Nr}_n{\sf CA}_{n+3}\) is not first order definable, where \(\mathbf{S_d}\) is the operation of forming dense subalgebras, and that for any \(2<n<m\), any \(l\geq n+3\) any any class \(\mathbf{K}\) (such that \({\sf At}({\sf Nr}_n{\sf CA}_{m})\cap {\sf CRCA}_n\subseteq \mathbf{K}\subseteq {\sf At}\mathbf{S_c}{\sf Nr}_n{\sf CA}_{l}\), \(\mathbf{K}\) is not not first order definable either.