# Definition:Lower Closure

## Contents

## Definition

### Lower Closure of an Element

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 $a$.

### Lower Closure of a Set

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

Let $T \subseteq S$.

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

- $T^\preccurlyeq := \bigcup \left\{{t^\preccurlyeq: t \in T}\right\}$

where $t^\preccurlyeq$ is the lower closure of $t$.

That is:

- $T^\preccurlyeq := \left\{{l \in S: \exists t \in T: l \preccurlyeq t}\right\}$

## Notation

On $\mathsf{Pr} \infty \mathsf{fWiki}$ we employ the following notational conventions for the closure operators on $\left({S, \preccurlyeq}\right)$ of an element $a$ of $S$.

- $a^\preccurlyeq := \left\{{b \in S: b \preccurlyeq a}\right\}$: the lower closure of $a \in S$: everything in $S$ that precedes $a$

- $a^\succcurlyeq := \left\{{b \in S: a \preccurlyeq b}\right\}$: the upper closure of $a \in S$: everything in $S$ that succeeds $a$

- $a^\prec := \left\{{b \in S: b \preccurlyeq a \land a \ne b}\right\}$: the strict lower closure of $a \in S$: everything in $S$ that strictly precedes $a$

- $a^\succ := \left\{{b \in S: a \preccurlyeq b \land a \ne b}\right\}$: the strict upper closure of $a \in S$: everything in $S$ that strictly succeeds $a$.

Similarly for the closure operators on $\left({S, \preccurlyeq}\right)$ of a subset $T$ of $S$:

- $\displaystyle T^\preccurlyeq := \bigcup \left\{{t^\preccurlyeq: t \in T:}\right\}$: the lower closure of $T \in S$: everything in $S$ that precedes some element of $T$

- $\displaystyle T^\succcurlyeq := \bigcup \left\{{t^\succcurlyeq: t \in T:}\right\}$: the upper closure of $T \in S$: everything in $S$ that succeeds some element of $T$

- $\displaystyle T^\prec := \bigcup \left\{{t^\prec: t \in T:}\right\}$: the strict lower closure of $T \in S$: everything in $S$ that strictly precedes some element of $T$

- $\displaystyle T^\succ := \bigcup \left\{{t^\succ: t \in T:}\right\}$: the strict upper closure of $T \in S$: everything in $S$ that strictly succeeds some element of $T$.

The astute reader may point out that, for example, $a^\preccurlyeq$ is ambiguous as to whether it means:

- The lower closure of $a$ with respect to $\preccurlyeq$
- The upper closure of $a$ with respect to the dual ordering $\succcurlyeq$

By Lower Closure is Dual to Upper Closure and Strict Lower Closure is Dual to Strict Upper Closure, the two are seen to be equal.

### Also denoted as

Other notations for closure operators include:

- ${\downarrow} a, {\bar \downarrow} a$ for lower closure of $a \in S$

- ${\uparrow} a, {\bar \uparrow} a$ for upper closure of $a \in S$

- ${\downarrow} a, {\dot \downarrow} a$ for strict lower closure of $a \in S$

- ${\uparrow} a, {\dot \uparrow} a$ for strict upper closure of $a \in S$

and similar for upper closure, lower closure, strict upper closure and strict lower closure of a subset.

However, as there is considerable inconsistency in the literature as to exactly which of these arrow notations is being used at any one time, its use is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also known as

A **lower closure** can also be referred to as a **weak lower closure** to distinguish it from a **strict lower closure**.