Definition:Recursively Defined Mapping

Definition

Natural Numbers

Let $p \in \N$ be a natural number.

Let $S = \set {x \in \N: p \le x}$.

Let $T$ be a set.

Let $g: T \to T$ be a mapping.

Let $f: S \to T$ be the mapping defined as:

$\forall x \in S: \map f x = \begin{cases} a & : x = p \\ \map g {\map f n} & : x = n + 1 \end{cases}$

for $a \in T$.

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

Peano Structure

Let $\struct {P, 0, s}$ be a Peano structure.

Let $T$ be a set.

Let $g: T \to T$ be a mapping.

Let $f: P \to T$ be the mapping defined as:

$\forall x \in P: \map f x = \begin{cases} a & : x = 0 \\ \map g {\map f n} & : x = \map s n \end{cases}$

where $a \in T$.

Then $f$ is said to be recursively defined on $P$.

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 $S'$.

Minimally Inductive Set

Let $\omega$ be the minimally inductive set.

Let $T$ be a set.

Let $a \in T$.

Let $g: T \to T$ be a mapping.

Let $f: \omega \to T$ be the mapping defined as:

$\forall x \in \omega: \map f x = \begin{cases} a & : x = 0 \\ \map g {\map f n} & : x = n^+ \end{cases}$

where $n^+$ is the successor set of $n$.

Then $f$ is said to be recursively defined on $\omega$.