I'm trying to show that if $(B, i)$ is the (BA) completion of any partial order $P$ and $A$ is a complete subalgebra of $B$, then $i^{-1}[A]$ is a complete suborder of $P$.
Pure hunch says it's true, but i'm stuck at whether a complete subalgebra $A$ of a complete boolean $B$ algebra always intersect all dense subsets of $B$.
Thanks in advance!