Varieties of Two-Dimensional Diagonal-Free Cylindric Algebras. Part I.
Nick Bezhanishvili
Abstract:
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