Galois connection
If is a function, then for any subset of we can form the image and for any subset of we can form the inverse image Then and form a monotone Galois connection between the power set of and the power set of , both ordered by inclusion ⊆. There is a further adjoint pair in this situation: for a subset of , define Then and form a monotone Galois connection between the power set of and the power set of . In the first Galois connection, is the upper adjoint, while in the second Galois connection it serves as the lower adjoint.