\begin{definition}[Definition:Recursively Defined Mapping/Naturally Ordered Semigroup]
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Let $p \in S$.
Let $S' = \set {x \in S: p \preceq x}$.
Let $T$ be a set.
Let $g: T \to T$ be a mapping.
Let $f: S' \to T$ be the mapping defined as:
:$\forall n \in S': \map f x = \begin{cases}
a & : x = p \\
\map g {\map f n} & : x = n \circ 1
\end{cases}$
where $a \in T$.
Then $f$ is said to be '''recursively defined''' on $S'$.
\end{definition}