Question on complete bounded distributive lattices

Question

Suppose you have any complete bounded distributive lattice of the form $\langle \mathbf{A}, \wedge, \vee, 0,1 \rangle$. Now, suppose you have two $a,b \in \mathbf{A} $ such that $(a\wedge b) = 0$ where $a \neq 0$ and $b \neq 0$. Is it possible to find two $c,d \in \mathbf{A}$ such that $a \leq c < 1$, $b\leq d < 1$ and $(c \wedge d) \neq 0$.

Answer 1

Let $a=(0,1,1), b=(1,0,0), c=a, d=(1,0,1)$, which satisfies all your prescriptions, and we obviously have $(c \land d) \neq 0$