Varieties of two-dimensional cylindric algebras II

Abstract

In this chapter we survey recent developments in the theory of twodimensional cylindric algebras. In particular, we will deal with varieties of two-dimensional diagonal-free cylindric algebras and with varieties of two-dimensional cylindric algebras with the diagonal. It is well known that two-dimensional diagonal-free cylindric algebras correspond to the two variable equality-free fragment of classical first-order logic FOL, whereas two-dimensional cylindric algebras with the diagonal correspond to the two variable fragment of FOL with equality. It is also well known that one-dimensional cylindric algebras, also called Halmos monadic algebras, provide algebraic completeness for the one variable fragment of FOL. For a systematic discussion on the connection between various fragments of FOL and classes of (cylindric) algebras that correspond to these fragments we refer to [And-Nem-Sai,01].