Complete Representations and Neat Embeddings

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