First-order axiomatisations of representable relation algebras need formulas of unbounded quantifier depth

TL;DR

This paper demonstrates that the class of all representable relation algebras cannot be characterized by any first-order axioms with bounded quantifier depth, nor can their atom structures be finitely defined with limited variables.

Contribution

It proves the necessity of unbounded quantifier depth and infinite variables for first-order axiomatizations of representable relation algebras.

Findings

RRA cannot be axiomatized by bounded quantifier depth.

Atom structures of RRA cannot be finitely defined with limited variables.

Unbounded quantifier depth is essential for first-order axiomatizations.

Abstract

Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.