Strongly representable atom structures
Using constructions of Hirsch and Hodkinson, we show that the class of strongly atom structures for various cylindric-like algebras is not elementary. This applies to diagonal free reducts and polyadic algebras with and without equality. This follows from the simple observation, that the cylindric atom structures of dimension n, constructed by Hirsch and Hodkinson, are built from algebras that can be easily expanded to polyadic equality algebras, and that such expansions are generated by elements whose dimension sets <n.