operationsaddition

Operationsaddition is a term used in some mathematical and computer science contexts to denote the operation of combining two operations into a single operation. When the target codomain supports addition, the sum of two operations f and g is defined pointwise by (f+g)(x) = f(x) + g(x). This construction is standard for real-valued or vector-valued functions, and it extends to higher-arity operations and to function spaces.

Formally, let f and g be functions from a common domain X to a codomain Y that is closed under an addition operation. The sum f+g is the function that maps each input to the additive combination of the outputs. If Y is a commutative monoid with additive identity 0, then the set of all such functions forms a commutative monoid under addition, with 0 as the identity element.

Key properties include commutativity, (f+g) = (g+f); associativity, (f+g)+h = f+(g+h); and the existence of a zero operation that maps every input to 0. Distributivity with respect to composition is not automatic and depends on the specific structures involved.

Examples include real-valued functions on a domain, where (f+g)(x) = f(x) + g(x); or vector-valued functions where addition is performed componentwise.

Applications of operationsaddition appear in functional analysis, signal processing, and the design of algebraic frameworks for programming languages, where combining sub-operations into a single aggregate operation is useful. It is a fundamental construction in many areas of algebra and analysis and does not denote a separate, widely specialized object beyond standard addition of functions.

See also: addition, binary operation, function space, pointwise addition, semiring, monoid.