Blowing up and blurring finite Monk and rainbow algebras

TL;DR

This paper introduces new algebraic constructions using Monk-like and rainbow algebras to demonstrate that certain classes of relation and cylindric algebras are not elementary and that some classes are not atom canonical, employing graph-based methods.

Contribution

It provides a novel proof that classes of strongly representable relation algebras and finite dimensional cylindric algebras are non-elementary, and shows that SNr_n ext{CA}_{n+k} is not atom canonical, using graph-based constructions.

Findings

Classes of strongly representable relation algebras are not elementary.

Finite dimensional cylindric algebras of dimension >2 are not elementary.

SNr_n ext{CA}_{n+k} is not atom canonical for k ≥ 4.

Abstract

We use Monk like algebras to give a new proof that the classes of strongly representable relation algebras and finite dimensional cylindric algebras of dimension >2 are not elementary. Our construction is based on relation algebras have cylindric basis, so that we obtain the result for both in one go, since they are both based on the same graph. The proof also uses Erdos' graphs. We also show using a rainbow construction that the class SNr_n\CA_{n+k} is not atom canonical, for any k\geq 4.