# Category:Lower Closures

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This category contains results about Lower Closures in the context of Order Theory.

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

Let $\left({S, \preccurlyeq}\right)$ be an ordered set.

Let $a \in S$.

The **lower closure of $a$ (in $S$)** is defined as:

- $a^\preccurlyeq := \left\{{b \in S: b \preccurlyeq a}\right\}$

That is, $a^\preccurlyeq$ is the set of all elements of $S$ that precede $a$.

## Pages in category "Lower Closures"

The following 21 pages are in this category, out of 21 total.

### L

### S

- Segment of Auxiliary Relation is Subset of Lower Closure
- Strict Lower Closure in Restricted Ordering
- Strict Lower Closure is Dual to Strict Upper Closure
- Strict Lower Closure is Lower Set
- Strict Lower Closure is Lower Set/Proof 1
- Strict Lower Closure is Lower Set/Proof 2
- Supremum of Lower Closure of Element
- Supremum of Lower Closure of Set