Galois connection
It can be shown that "f" is residuated if and only if there exists a (necessarily unique) monotone function "f" +: "B" → "A" such that "f"  "f" + ≤ idB and "f" +  "f" ≥ idA, where id is the identity function. The function "f" + is the residual of "f". A residuated function and its residual form a Galois connection under the (more recent) monotone definition of that concept, and for every (monotone) Galois connection the lower adjoint is residuated with the residual being the upper adjoint. Therefore, the notions of monotone Galois connection and residuated mapping essentially coincide.