Varieties of Two-Dimensional Cylindric Algebras

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].

The author is grateful to Ian Hodkinson for his comments on the earlier version of this chapter. Special thanks go to the referee for valuable suggestions and pointers to the literature that substantially improved the presentation of this chapter.