In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets.
This approach gives rise to notions of left and right division in a partially ordered magma, additionally endowing it with a quasigroup structure.
If B° denotes the dual order (opposite poset) to B then f : A → B is a residuated mapping if and only if there exists an f * such that f : A → B° and f *: B° → A form a Galois connection under the original antitone definition of this notion.
[3] If • : P × Q → R is a binary map and P, Q, and R are posets, then one may define residuation component-wise for the left and right translations, i.e. multiplication by a fixed element.
Matrix multiplication is defined in the usual manner with the "product" being a meet, and the "sum" a join.