Let $(P,\le)$ be a partially ordered set and $\preceq$ the Hoare pre-order on its subsets, i.e. for $X,Y\subseteq P$, $X\preceq Y$ iff $\forall x\in X:\exists y\in Y:x\le y$.
Let $\sim$ be the equivalence on the power set $2^P$ generated by $\preceq$, i.e. $X\sim Y$ iff $X\preceq Y$ and $Y\preceq X$. Let $M=2^P/\mathord\sim$ be the quotient, then $\preceq$ passes to a partial order, also denoted $\preceq$, on $M$.
Is $(M,\preceq)$ a complete lattice?