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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 ≤ f. 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 ↓{b} = { b' ∈ B : b' ≤ b }. In general the preimage under f of a principal down-set need not be a principal down-set. If it is, f is called residuated.
The notion of residuated map can be generalized to a binary operator via component-wise residuation. This approach gives rise to notions of left and right division in a partially ordered magma, additionally endowing it with a quasigroup structure.. A binary residuated map is usually not residuated as a unary map.