Aqilah: I don't think I understand your first question; can you rephrase? The empty set is a subset of every set. If our entire poset is the empty set, then there is no meet of the empty set in that poset, since there are no elements to be the meet.

By "start as far up as we can", I just mean that \\(\bigwedge (A \cup \\{a\\}) \subseteq\bigwedge A\\). So \\(\bigwedge \emptyset\\) had better be very "big", so as not to put any restrictions on \\(\bigwedge\\{a\\}\\).