In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function.
If A, B are posets, a function f: A → B is defined to be monotone if it is order-preserving: that is, if x ≤ y implies f(x) ≤ f(y). This is equivalent to the condition that the preimage under f of every down-set of B is a down-set of A. We define a principal down-set to be one of the form ↓ =