funkci

Funkci is a term used in theoretical discussions of higher-order functions to denote a class of function-like constructs that are closed under composition and support an evaluation operation. In this sense, a funkci behaves similarly to a function, but is treated as a first-class object that can be combined and applied within a formal framework.

Formal definition (in the most common formulation): let D be a nonempty set. A funkci on D is a collection F of endomorphisms from D to itself such that F is closed under composition and contains the identity function id_D. The pair (D, F) provides a structure in which elements of F can be applied to elements of D, and the composition of two funkci yields another funkci in F.

Relation to other concepts: This structure corresponds to a monoid under the operation of composition, and, in category theory terms, a submonoid of End(D). In programming language semantics, funkci are sometimes used to illustrate higher-order function composition and evaluation strategies, such as partial application or currying.

Examples: If D is any set, the full endomorphism set End(D) forms a funkci. More elaborate funkci may restrict to monotone, continuous, or invertible endomorphisms depending on context. Some authors allow funkci to vary across different input types in a polymorphic setting.

History and usage: The term funkci is not universally standardized. It appears in informal notes and introductory expositions as a convenient label for function-like composing structures. As such, precise properties of a funkci may differ between texts; readers should check the given definitions in each source.

See also: Function, Higher-order function, Function composition, Monoid, Endomorphism, Category theory.