Strongly representable atom structures

TL;DR

This paper demonstrates that the class of strongly atom structures for various cylindric-like algebras is not elementary, using constructions by Hirsch and Hodkinson to show limitations in their axiomatizability.

Contribution

It shows that the class of strongly atom structures for cylindric-like algebras is non-elementary, extending previous constructions to a broader class of algebras.

Findings

The class of strongly atom structures is not elementary.

Hirsch and Hodkinson's constructions can be expanded to polyadic equality algebras.

Such expansions are generated by elements with dimension sets less than n.

Abstract

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.