# PP-2000-10: Varieties of Two-Dimensional Diagonal-Free Cylindric Algebras. Part I.

PP-2000-10: Bezhanishvili, Nick (2000) Varieties of Two-Dimensional Diagonal-Free Cylindric Algebras. Part I. [Report]

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This is the first part of the whole work which will consist of two parts and intends to obtain a clear picture of the lattice $\Lambda({\bf Df}_2)$ of all subvarieties of the variety {\bf Df}$_2$ of the two-dimensional diagonal-free cylindric algebras. Here we show that every proper subvariety of {\bf Df}$_2$ is locally finite, and hence {\bf Df}$_2$ is hereditarily finitely approximable. Moreover, we prove that there exist exactly six critical varieties in $\Lambda({\bf Df}_2)$, and characterize finite subvarieties of {\bf Df}$_2$. It is also shown that a variety ${\bf V}\in\Lambda({\bf Df}_2)$ is representable by its square algebras iff either ${\bf V}={\bf Df}_2$ or {\bf V} is a finite variety, and give a necessary and sufficient condition for a finite variety to be representable. Representable varieties by their rectangular algebras are also described. The complexity of $\Lambda({\bf Df}_2)$ will be investigated in Part II. Keyword(s): Cylindric Algebra Theory