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
.