# Definition:Successor

## Disambiguation

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**Successor** may refer to:

- Successor element: In an ordered set $\left({S, \preceq}\right)$, $a$ is a successor element to $b$ if and only if $b \prec a$.

- Immediate successor element: In an ordered set $\left({S, \preceq}\right)$, $a$ is the immediate successor element to $b$ if and only if $b \prec a$ and $\nexists c \in S: b \prec c \prec a$.

- Successor set: If $S$ is a set, then its successor set $S^+$ is defined as $S^+ := S \cup \left\{{S}\right\}$.

- Successor mapping: The mapping at the heart of a Peano structure which encapsulates its ability to sustain the Principle of Mathematical Induction.

- Successor ordinal: An ordinal which is the successor set of some other ordinal.