There is no finite variable axiomatization for various diagonal free algebras

TL;DR

This paper proves that for certain classes of algebraic structures related to logic, no finite set of axioms using a limited number of variables can characterize all representable algebras, solving a longstanding open problem.

Contribution

It demonstrates the non-existence of finite variable universal axiomatizations for classes between diagonal free cylindric algebras and polyadic equality algebras, resolving a 1990 open problem.

Findings

No finite variable universal axiomatization exists for these algebraic classes.

The result applies to classes of representable algebras between specific algebraic structures.

The proof uses a rainbow construction for cylindric 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.