Definition:Additive Function (Conventional)

This page is about functions preserving addition. For other uses, see Definition:Additive Function.

Definition

Let $f: S \to S$ be a mapping on an algebraic structure $\left({S, +}\right)$.

Then $f$ is an additive function iff it preserves the addition operation:

$\forall x, y \in S: f \left({x + y}\right) = f \left({x}\right) + f \left({y}\right)$

Examples

In the field of abstract algebra, this operation can be seen to be a endomorphism on $\left({S, +}\right)$.

In the field of linear algebra, it can be seen that a linear transformation is additive.

When the domain is the set of real numbers, this is the Cauchy Functional Equation.