  For a finite-type star operation $\star$ on a domain $R$, we say that $R$ is
$\star$-super potent if each maximal $\star$-ideal of $R$ contains a finitely
generated ideal $I$ such that (1) $I$ is contained in no other maximal
$\star$-ideal of $R$ and (2) $J$ is $\star$-invertible for every finitely
generated ideal $J \supseteq I$. Examples of $t$-super potent domains include
domains each of whose maximal $t$-ideals is $t$-invertible (e.g., Krull
domains). We show that if the domain $R$ is $\star$-super potent for some
finite-type star operation $\star$, then $R$ is $t$-super potent, we study
$t$-super potency in polynomial rings and pullbacks, and we prove that a domain
$R$ is a generalized Krull domain if and only if it is $% t $-super potent and
has $t$-dimension one.
