Strongly representable algebras

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 Andréka 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 $\A\in \Nr_n\CA_ω$. We could prove one side.

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