# Definition:Minimally Inductive Class under General Mapping

## Definition

Let $A$ be a class.

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

### Definition 1

$A$ is minimally inductive under $g$ if and only if:

 $(1)$ $:$ $A$ is inductive under $g$ $(2)$ $:$ No proper subclass of $A$ is inductive under $g$.

### Definition 2

$A$ is minimally inductive under $g$ if and only if:

 $(1)$ $:$ $A$ is inductive under $g$ $(2)$ $:$ Every subclass of $A$ which is inductive under $g$ contains all the elements of $A$.

### Definition 3

$A$ is minimally inductive under $g$ if and only if $A$ is minimally closed under $g$ with respect to $\O$.

## Also see

• Results about minimally inductive classes can be found here.