# Category:Lower Sections

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This category contains results about **Lower Sections**.

Definitions specific to this category can be found in **Definitions/Lower Sections**.

Let $\struct {S, \preceq}$ be an ordered set.

Let $L \subseteq S$.

### Definition 1

$L$ is a **lower section** in $S$ if and only if:

- $\forall l \in L, s \in S: s \preceq l \implies s \in L$

### Definition 2

$L$ is a **lower section** in $S$ if and only if:

- $L^\preceq \subseteq L$

where $L^\preceq$ is the lower closure of $L$.

### Definition 3

$L$ is a **lower section** in $S$ if and only if:

- $L^\preceq = L$

where $L^\preceq$ is the lower closure of $L$.

## Subcategories

This category has only the following subcategory.

## Pages in category "Lower Sections"

The following 22 pages are in this category, out of 22 total.