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}.

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