Definition:Recursively Defined Mapping/Naturally Ordered Semigroup

Definition
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.

Let $p \in S$.

Let $S' = \left\{{x \in S: p \preceq x}\right\}$.

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': f \left({x}\right) = \begin{cases}

a & : x = p \\ g \left({f \left({n}\right)}\right) & : x = n \circ 1 \end{cases}$

where $a \in T$.

Then $f$ is said to be recursively defined on $S'$.

Also see

 * Principle of Recursive Definition/General Result