Definition:Successor


 * Successor element: In a poset $\left({S, \preceq}\right)$, $a$ is a successor element to $b$ iff $b \prec a$.


 * Immediate successor element: In a poset $\left({S, \preceq}\right)$, $a$ is the immediate successor element to $b$ iff $b \prec a$ 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 Finite Induction.


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