Characterizing injective homomorphisms on injective objects in the category of boolean algebras

Question

I'm trying to understand what type of injective homomorphism can have as domain an injective object in the category of boolean algebras. I have in mind two simple types of homomorphisms with domain a boolen algebra: the splitting ones and the complete ones. Do the two notions overlap when the domain is a complete boolean algebra?

Recall that $i$ splitting means that there is $j:C\to B$ homomorphism of boolean algebras such that $j\circ i=Id_B$.

Assume $i:B\to C$ is a splitting injective homomorphism of boolean algebras with $B$ complete. Is $i$ necessarily a complete homomorphism?

The converse holds. Assume $i:B\to C$ is a complete injective homomorphism of boolean algebras with $B$ complete. Then $i$ is splitting just because complete boolean algebras are the injective objects in the category of boolean algebras (we just need that $i$ is injective with domain a cba to have its splitting decomposition, not that $i$ is complete).

So the question really amounts to ask if injective homomorphism of booelan algebras wih domain a complete boolean algebra are complete homomorphisms.

Answer 1

This is not an answer but since the question didn't get so much attention, I think this comment might be useful to work further on it.

The aim is to prove or disprove that if $\alpha: B \to C$ is a monomorphism between Boolean algebras and $B$ is complete, then $\alpha$ preserves arbitrary joins. Since joins are colimits when we think of $B$ as a category, and since left adjoints preserve colimits, it would be enough to prove that $\alpha$ has a right adjoint.

Defining $\beta: C \to B$ to be $$\beta(c):=\bigvee\{b \in B: \alpha(b) \le c\}$$ doesn't always work. $\beta$ is well-defined (since $B$ is complete), order-preserving and \begin{equation} (1) \quad\alpha(b) \le c \quad \implies \quad b \le \beta(c)\text{.} \end{equation}

The other direction is trickier. For example, as @curious on mathematics suggests, if $\alpha$ is the quotient map $\nu : \mathcal{P}(\mathbb{N}) \to \mathcal{P}(\mathbb{N})/ I$, where is the ideal of finite sets, then $\beta$ does not satisfy the opposite direction in (1).