Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

Abstract.

We investigate the lattice $\Lambda (\textbf{Df}_2)$ of all subvarieties of the variety Df ₂ of two-dimensional diagonal-free cylindric algebras. We prove that a Df ₂-algebra is finitely representable if it is finitely approximable, characterize finite projective Df ₂-algebras, and show that there are no non-trivial injectives and absolute retracts in Df ₂. We prove that every proper subvariety of Df ₂ is locally finite, and hence Df ₂ is hereditarily finitely approximable. We describe all six critical varieties in $\Lambda (\textbf{Df}_2)$, which leads to a characterization of finitely generated subvarieties of Df ₂. Finally, we describe all square representable and rectangularly representable subvarieties of Df ₂.