# Definition:Successor Mapping

## Definition

Let $\left({P, s, 0}\right)$ be a Peano structure.

Then the mapping $s: P \to P$ is called the **successor mapping on $P$**.

The image element $s \left({x}\right)$ of an element $x$ is called the **successor element** or just **successor** of $x$.

### Successor Mapping on Natural Numbers

Let $\N$ be the set of natural numbers.

Let $s: \N \to \N$ be the mapping defined as:

- $s = \left\{{\left({x, y}\right): x \in \N, y = x + 1}\right\}$

Considering $\N$ defined as a Peano structure, this is seen to be an instance of a successor mapping.

## Also known as

The **successor mapping** can also be seen referred to as the **successor function**.

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*... (previous) ... (next): Introduction $\S 4$: The natural numbers - 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 1.1$: Introduction - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 16$