saturatedMapping

A saturatedMapping is a function f: X → Y defined with respect to an equivalence relation ~ on X. It is characterized by the property that for every y in Y, the fiber f^{-1}({y}) is saturated with respect to ~; equivalently, f is constant on each equivalence class of ~. In other words, there exists a unique map g from the quotient set X/~ to Y such that f = g ∘ q, where q: X → X/~ is the quotient map.

Saturated fibers and invariance are central: if x ~ x', then f(x) = f(x'). This ensures that f depends only on the class of x, not on the particular representative. The equivalence between the two viewpoints is that a function is saturated exactly when it factors through the quotient X/~, yielding a well-defined induced map on the quotient.

Properties of saturatedMappings include stability under composition with post- and pre-composition: if h: Y → Z is any function, then h ∘ f is saturated, since (h ∘ f)^{-1}({z}) = f^{-1}(h^{-1}({z})), a union of f^{-1}({y}) fibers, each saturated. The collection of saturatedMappings from X to Y is naturally in bijection with all functions from the quotient X/~ to Y.

Examples illustrate the idea: let X = Z with ~ defined by congruence modulo n. A function f that depends only on the residue class, such as f(x) = x mod n, is a saturatedMapping because it is invariant on each class and factors through Z/nZ.

Applications of saturatedMappings appear in quotient constructions, data aggregation by equivalence classes, and various areas of mathematics where invariance under a partition plays a role. See also quotient sets, equivalence relations, and invariance.