In mathematics, the concept of a residuated mapping arises in the theory ofpartially 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 ↓ =. 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 orderedmagma, additionally endowing it with a quasigroupstructure.. A binary residuated map is usually not residuated as a unary map.
Definition
If A, B are posets, a function f: A → B is residuatedif and only if the preimage under f of every principal down-set of B is a principal down-set of A.
Consequences
With A, B posets, the set of functionsA → B can be ordered by the pointwise orderf ≤ g ↔ f ≤ g. It can be shown that f is residuated if and only if there exists a monotone function f +: B → A such that f o f + ≤ idB and f + o 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 monotone definition of that concept, and for every 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. Additionally, we have f -1 = ↓. If B° denotes the dual order 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. If f : A → B and g : B → C are residuated mappings, then so is the function compositionfg : A → C, with residual + = g +f +. The antitone Galois connections do not share this property. The set of monotone transformations over a poset is an ordered monoid with the pointwise order, and so is the set of residuated transformations.
Examples
The ceiling function from R to Z is residuated, with residual mapping the natural embedding of Z into R.
The embedding of Z into R is also residuated. Its residual is the floor function.
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. For an element x in P define xλ = x • y, and for x in Q define λx = y • x. Then • is said to be residuated if and only if xλ and λx are residuated for all x. Left division are defined by taking the residuals of the left translations: x\y = + and x/y = + For example, every ordered group is residuated, and the division defined by the above coincides with notion of division in a group. A less trivial example is the set Matn of square matrices over a boolean algebraB, where the matrices are ordered pointwise. The pointwise order endows Matn with pointwise meets, joins and complements. Matrix multiplication is defined in the usual manner with the "product" being a meet, and the "sum" a join. It can be shown that X\Y = ' and X/Y =.
