Compositeeach

Compositeeach is a concept in computer science and data processing that describes a rule for applying a function to every atomic element contained within a composite data structure, while preserving the structure of the outer container. The term combines composite, referencing data structures such as lists of lists, trees, or graphs, with each, indicating an element-wise operation.

Definition: For a data structure C that contains leaves of type A, a compositeeach operation takes a function f: A -> B and returns a new structure C with the same topology but leaves transformed to B. This generalizes map for single-level containers and can compose with other operations to traverse nested structures.

Properties: It is structure-preserving and can be referentially transparent if f is. It is compatible with composition and can be implemented via recursion, higher-order functions, or specialized traversals. In categorical terms, compositeeach can be viewed as a functorial action across a container, enabling uniform element-wise processing regardless of nesting depth.

Examples: Applying compositeeach to [[1,2],[3,[4]]] with f(x) = x+1 yields [[2,3],[4,[5]]]. In a binary tree, applying a leaf function to all values yields a tree of results with the same shape.

Applications: Data transformation pipelines, schema migrations, UI component trees, serialization, and validation tasks that require uniform leaf processing without altering structure.

Implementation notes: Real-world use requires careful handling of mixed-type leaves, error propagation, and performance considerations for large or lazy structures.

See also: map, traverse, functor, applicative, recursion schemes, tree traversal.