functionalthatf

Functionalthatf is a theoretical construct used in discussions of higher-order functions and functional programming semantics. It denotes a class of higher-order operators that take a function as input and return another function according to a fixed rule. In formal terms, functionalthatf can be described as a family of maps that act on endofunctions, transforming one function into another within the same function space.

Definition: Let X be a nonempty set. For each fixed endomap s: X → X, define F_s: End(X) → End(X), where End(X) is the set of all functions from X to X, by a deterministic rule that depends on s. The collection {F_s} is referred to as the functionalthatf family. Common instantiations include precomposition (F_s(f) = f ∘ s), postcomposition (F_s(f) = s ∘ f), and iteration (F(f) = f ∘ f).

Properties: Functionalthatf operators are typically described as referentially transparent, meaning F_s(f) is determined solely by f and s. They are not required to preserve composition in general, unless a particular rule imposes such a property. The framework is often used to discuss how higher-order transformations interact with function spaces and to illustrate ideas about functor-like behavior in a simplified setting.

Examples:

- Precomposition: F_s(f) = f ∘ s.

- Postcomposition: F_s(f) = s ∘ f.

- Iteration: F(f) = f ∘ f.

Usage and context: The term functionalthatf is not a standard object in mainstream mathematics but serves as a neutral placeholder for describing families of function-transforming operators in expository or theoretical discussions, including topics in higher-order programming, program transformation, and category-inspired perspectives on function spaces.

See also: higher-order function, endofunction, composition, functor, functional programming.