Definition:Successor Mapping

Definition

Let $\struct {P, s, 0}$ be a Peano structure.

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

The image element $\map s x$ 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.