# Category:Definitions/Minimally Closed Classes

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This category contains definitions related to Minimally Closed Classes.

Related results can be found in Category:Minimally Closed Classes.

Let $A$ be a class.

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

### Definition 1

$A$ is **minimally closed under $g$ with respect to $b$** if and only if:

\((1)\) | $:$ | $A$ is closed under $g$ | ||||||

\((2)\) | $:$ | There exists $b \in A$ such that no proper subclass of $A$ containing $b$ is closed under $g$. |

### Definition 2

$A$ is **minimally closed under $g$ with respect to $b$** if and only if:

\((1)\) | $:$ | $A$ is closed under $g$ | ||||||

\((2)\) | $:$ | There exists $b \in A$ such that every subclass of $A$ containing $b$ which is closed under $g$ contains all the elements of $A$. |

## Subcategories

This category has only the following subcategory.

## Pages in category "Definitions/Minimally Closed Classes"

The following 3 pages are in this category, out of 3 total.