# 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 $\mathfrak{A}$ and $\mathfrak{B}$ be complete boolean lattices. Then $\alpha$ 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 $\beta$ the upper adjoint of $\alpha$ if $(\alpha;\beta)$ is a boolean funcoid? (Equivalently: Is $(\alpha;\beta)$ a boolean funcoid if $(\alpha;\beta)$ is a Galois connection between complete boolean lattices $A$ and $B$?)

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