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 Df2 of two-dimensional diagonal-free cylindric algebras. We prove that a Df2-algebra is finitely representable if it is finitely approximable, characterize finite projective Df2-algebras, and show that there are no non-trivial injectives and absolute retracts in Df2. We prove that every proper subvariety of Df2 is locally finite, and hence Df2 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 Df2. Finally, we describe all square representable and rectangularly representable subvarieties of Df2.