Definition:Minimally Inductive Class under General Mapping

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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.


Sources