Definition:Recursively Defined Mapping/Naturally Ordered Semigroup

Definition

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 $S'$.

Also see

• Principle of Recursive Definition/General Result

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Theorem $16.6$