Strongly representable algebras

TL;DR

This paper provides a simplified proof of a result on the non-elementarity of strongly representable atom structures, using blow-up and blur constructions, and explores implications for various algebra classes.

Contribution

It introduces a simplified proof technique for strong representability results and extends the analysis to classes between diagonal free and polyadic algebras, including infinite dimensions.

Findings

Strongly representable atom structures are not elementary.

A blow-up and blur construction demonstrates the non-elementarity.

The paper proposes a potential equivalence involving representability and neat reducts.

Abstract

We give a simpler proof of a result of Hodkinson in the context of a blow and blur up construction argueing that the idea at heart is similar to that adopted by Andr\'eka et all \cite{sayed}. The idea is to blow up a finite structure, replacing each 'colour or atom' by infinitely many, using blurs to represent the resulting term algebra, but the blurs are not enough to blur the structure of the finite structure in the complex algebra. A reverse of this process exists in the literature, it builds algebras with infinite blurs converging to one with finite blurs. This idea due to Hirsch and Hodkinson, uses probabilistic methods of Erdos to construct a sequence of graphs with infinite chromatic number one that is 2 colourable. This construction, which works for both relation and cylindric algebras, further shows that the class of strongly representable atom structures is not elementary. We will generalize such a result for any class of algebras between diagonal free algebras and polyadic algebras with and without equality, then we further discuss possibilities for the infinite dimensional case. Finally, we suggest a very plausible equivalence, and that is: If $n>2$, is finite, and $\A\in \CA_n$ is countable and atomic, then $\Cm\At\A$ is representable if and only if $\A\in \Nr_n\CA_{\omega}$. We could prove one side.