Let $\mathbb{B} = \langle B, \wedge, \vee, \leq, \neg, 0, 1 \rangle$ be an atomless complete Boolean algebra.

We call $f$ a splitting function on $\mathbb{B}$ iff

1. $f : B-\{1\} \longrightarrow (B-\{1\} \times B-\{1\}) \ \ \ (b \mapsto (b_0, b_1))$,
2. $b_i \leq b$ for all $b \in B-\{1\}$ and $i \in \{0, 1\}$,
3. $b_0 \wedge b_1 = 0$ for all $b \in B-\{1\}$.

A splitting function on $\mathbb{B}$ is monotone iff

4. $b \leq b'$ implies $b_i \leq b_{i}'$, for all $\{b, b'\} \subset B-\{1\}$ and $i \in \{0, 1\}$.

Monotone splitting functions trivially exist: consider eg. the map $b \mapsto (0, b)$. However, it is not so clear to me how non-trivial these can be.

In particular, is there always a monotone splitting function on $\mathbb{B}$ satisfying

5. $sup\{b_{i}' : b \leq b'\} = \neg b_{i-1}$ for all $b \in B-\{1\}$ and $i \in \{0, 1\}$?

If not, what additional conditions on $\mathbb{B}$ can guarantee its existence, and in what cases does it fail to exist?