Conversely, suppose that L possesses a dense element, say d. So [d]∗ = (0].
Clearly, [d]∗ ∈ N0(L). Now for any x ∈ L, consider [x]∗ ∩ [d]∗ = [x]∗ ∩ (0] =
[x]∗ ∩(0] = (0]. Also [x]∗∨ [d]∗ = {[x]∗∗ ∩[d]∗∗}∗ = {[x]∗∗ ∩[0]∗}∗ = {[x]∗∗ ∩L}∗ =
[x]∗∗∗ = [x]∗. Hence [d]∗ is the smallest element in N0(L).