# Category:Minimally Inductive Classes

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This category contains results about **Minimally Inductive Classes**.

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 $\O$.

## Subcategories

This category has the following 7 subcategories, out of 7 total.

### M

### N

## Pages in category "Minimally Inductive Classes"

The following 13 pages are in this category, out of 13 total.

### M

- Minimally Inductive Class under Progressing Mapping induces Nest
- Minimally Inductive Class under Progressing Mapping is Well-Ordered under Subset Relation
- Minimally Inductive Class under Progressing Mapping with Fixed Element is Finite
- Minimally Inductive Class under Slowly Progressing Mapping is Nest