There is no finite variable axiomatization for various diagonal free algebras

Abstract

We show, using a ranbow construction for cylindric algebras, that for any class K between diagonal free cylindric algebras and polyadic equality algebras of finite dimension > 2, there is no finite variable universal axiomatization for the class of representable algebras. This solves an old open problem in algebraic logic, formulated by Sain and Thompson back in 1990.