  We show that quasi-projective relation algebras and directed cylindric
algebras are equivalent categorialy. We work out a Godels second incompleteness
theorem for finite varibale fragments of first order logic. We show that
distinct set theories (like one with CH, and another with its negation) give
rise to equationally distinct simple directed cylindric algebras. This
correspondance was worked out for quasi projective relation algebras (with a
distinguished element corresponding to membership relation). The idea of the
proof is that the Sagi representation of directed cylindric algebras preserve
well-foundnes. Finally, using that the functor defined preserves order, we show
that the class of directed cylindric algebras have the superamalgmation
property.
