I call pointfree funcoids (see my free e-book) between boolean lattices as boolean funcoids.
I have proved that:
Theorem Let and be complete boolean lattices. Then 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 the upper adjoint of if is a boolean funcoid? (Equivalently: Is a boolean funcoid if is a Galois connection between complete boolean lattices and ?)
Further idea: We can define pointfree reloids between posets and as filters on the set of Galois connections between and .