Doesn't the Pareto-extension rule invalidate Eliaz's (2004) unified theorem of social choice?
Acyclicality as defined by Eliaz is simply the condition that
the values of a social aggregator are transitive. It is different from the usual acyclicity condition according to which a relation is acyclic iff it has irreflexive transitive closure. The formulation of Eliaz is a bit cryptic, but he actually shows in his Observation 1 that his acyclicality implies transitivity. The converse is also easy; the simple details are below. The result of the Pareto extension rule need of course not be transitive.
Formally, acyclicality as defined by Eliaz says that for all $R\in\mathbf{R}$ and for every three alternatives $a$, $b$ and $c$ in $A$, if $aRb$ and $\neg(cRb)$, then $\neg(cRa)$.
But the condition that for every three alternatives $a$, $b$ and $c$ in $A$, if $aRb$ and $\neg(cRb)$, then $\neg(cRa)$ is simply the transitivity of $R$, formulated in a slightly weird but equivalent way.
Here is the simple proof: Assume that $R$ is transitive and $aRb$ and $\neg(cRb)$. If it were the case that $cRa$, then $cRb$ by transitivity, contradicting the assumption. For the other direction, assume that the condition holds and that $cRa$ and $cRb$ holds. We want to show that $cRb$. Indeed, if not then by the condition $\neg(cRa)$, contradicting the assumption.