Galois connections are related with pointfree funcoids!

I call pointfree funcoids (see my free e-book) between boolean lattices as boolean funcoids.

I have proved that:

Theorem Let $latex \mathfrak{A}$ and $latex \mathfrak{B}$ be complete boolean lattices. Then $latex \alpha$ is the first component of a boolean funcoid iff it is a lower adjoint (in the sense of Galois connections between posets).

Does this theorem generalize for non-complete boolean lattices? or even further?

Is $latex \beta$ the upper adjoint of $latex \alpha$ if $latex (\alpha;\beta)$ is a boolean funcoid? (Equivalently: Is $latex (\alpha;\beta)$ a boolean funcoid if $latex (\alpha;\beta)$ is a Galois connection between complete boolean lattices $latex A$ and $latex B$?)

Further idea: We can define pointfree reloids between posets $latex \mathfrak{A}$ and $latex \mathfrak{B}$ as filters on the set of Galois connections between $latex \mathfrak{A}$ and $latex \mathfrak{B}$.