Name of poset operation involving Möbius function

Question

If we have a finitary poset $P$ with a $\hat{0}$ (least element), and we want to compute the Möbius function for all elements $x$, as in $\mu(\hat{0},x)$, it wouldn't affect any of our computations to delete elements $y$ with $\mu(\hat{0},y)=0$. It could affect computations of $\mu(x,z)$ for arbitrary $x,z$, but say we only care about the case when $x=\hat{0}$. Is there a name for this operation of deleting all such elements from a poset, or for a poset where all elements have nonzero Möbius function, e.g. the Boolean lattice?