Vacuous Reduct Semantics

Given a base semantics $\sigma$ and a reduct semantics $\tau$, a set $E \subseteq A$ is a $\sigma^{\tau}$-extension iff $E \in \sigma(F)$ and the reduct $F^E$ is $\tau$-vacuous, i.e. $\tau(F^E) \subseteq \{\emptyset\}$.