A cylindric algebra atom structure is said to be
strongly representable
if all atomic cylindric algebras with that atom structure are
representable. This is equivalent to saying that the full complex algebra
of the atom structure is a representable cylindric algebra. We show that for any finite
n≥3, the class of all strongly representable n-dimensional cylindric algebra atom
structures is not closed under ultraproducts and is therefore not
elementary.
Our proof is based on the following construction. From an arbitrary
undirected, loop-free graph Γ, we construct
an n-dimensional atom structure
ℰ(Γ), and prove, for infinite Γ, that ℰ(Γ)
is a strongly representable cylindric algebra atom structure if and only if the
chromatic number of Γ is infinite. A construction of Erdős
shows that there are graphs Γk (k < ω) with infinite
chromatic number, but having a non-principal ultraproduct
∏DΓk whose chromatic number is just two. It follows that
ℰ(Γk) is strongly representable (each k <ω) but
∏Dℰ(Γk) is not.
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