Metaoperation

Metaoperation is a term used to describe an operation that acts on other operations. It denotes a higher-order operation: an operation that takes one or more operations as input and yields a new operation or its result. The concept appears across mathematics and computer science as a way to reason about how processes, functions, or algebraic rules can be combined, transformed, or analyzed at a higher level.

Formally, a metaoperation can be viewed as a function whose inputs are operations of a given type. For example, function composition is a metaoperation that takes two functions and returns their composite function. Iteration, which applies a function repeatedly, is another metaoperation on the same function. In programming, higher-order functions such as map or fold act as metaoperations on other functions, producing new functions or results.

In algebra and category theory, metaoperations are closely related to the notion of composition rules in operads and higher-order morphisms. They provide a framework for describing how to compose, transform, or rearrange basic operations according to specified laws. This formal viewpoint supports meta-programming, language design, and the analysis of systems by treating operations themselves as first-class objects.

Because the term is used informally in various disciplines, its precise meaning can vary with context. Generally, metaoperation emphasizes higher-order structure—how operations relate to, combine, or manipulate other operations rather than their immediate outputs. Related concepts include higher-order functions, morphisms, and operads.