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

Rob Egrot, Robin Hirsch
2021 arXiv   pre-print
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.
arXiv:2008.01329v3 fatcat:xhbnj6w4ebdl7ftt2i5f7yirhm