Definition:Additive Function (Algebra)

This page is about additive function in the context of algebra. For other uses, see Additive Function.

Definition

Let $f: S \to S$ be a mapping on an algebraic structure $\struct {S, +}$.

Then $f$ is an additive function if and only if it preserves the addition operation:

$\forall x, y \in S: \map f {x + y} = \map f x + \map f y$

Examples

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

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.

Example: $\map f x = 3 x$

Let $\map f x$ be the real function defined as:

$\forall x \in \R: \map f x = 3 x$

Then $f$ is an additive function.

Square Root is not Additive

The square root function is not an additive function.

Also see

• Results about additive functions can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): additive: 1. (of a function between semigroups)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): additive function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): additive function
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): additive function
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): additive function