I'm interested in a particular class of maps between boolean algebras: the homomorphisms $f:A\to B$ such that there exist functions (not necessarily homomorphisms) $u,d:B\to A$ such that
$$f(a)\leq b\Leftrightarrow a\leq u(b)\text{ and }b\leq f(a)\Leftrightarrow d(b)\leq a$$
i.e. such that $d\dashv f\dashv u$ when seen as functors between posets-as-categories.
Have such homomorphisms been studied in the literature? What names are they known by?
I know that in topos theory there are similar concepts called atomic and essential geometric morphisms (for boolean algebras "atomic" and "essential" morphisms are the same, since if a function $f$ is any two of (homomorphism, left-adjoint, right-adjoint) then it is also the third).
I also know that if $A$ is complete then these homomorphisms are precisely the ones that preserve all meets and joins (by the adjoint functor theorem).