# PP-2002-18: Varieties of Two-Dimensional Cylindric Algebras. Part II

PP-2002-18: Bezhanishvili, Nick (2002) Varieties of Two-Dimensional Cylindric Algebras. Part II. [Report]

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In the precursor to this report, we investigated the lattice $\Lambda(Df_2)$ of all subvarieties of the variety $Df_2$ of two-dimensional diagonal free cylindric algebras. In the present paper we investigate the lattice $\Lambda(CA_2)$ of all subvarieties of the variety $CA_2$ of two-dimensional cylindric algebras. We give a dual characterization of representable two-dimensional cylindric algebras, prove that the cardinality of $\Lambda(CA_2)$ is that of continuum, give a criterion for a subvariety of $CA_2$ to be locally finite, and describe the only pre locally finite subvariety of $CA_2$. We also characterize finitely generated subvarieties of $CA_2$ by describing all fifteen pre finitely generated subvarieties of $CA_2$. Finally, we give a rough picture of $\Lambda(CA_2)$, and investigate algebraic properties preserved and reflected by the reduct functors $F : CA_2 \to Df_2$ and $\Phi : \Lamda(CA_2) \to \Lambda(Df2)$.