# Definition:Lower Closure

## Definition

### Lower Closure of Element

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

Let $a \in S$.

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

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

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

### Lower Closure of Subset

Let $\struct {S, \preccurlyeq}$ be an ordered set or preordered set.

Let $T \subseteq S$.

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

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

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

That is:

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

## Also known as

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

## Also defined as

Some sources define a **lower closure** with respect to a **total ordering** only, not considering the same concept with respect to a partial ordering.

## Notation

On $\mathsf{Pr} \infty \mathsf{fWiki}$ we employ the following notational conventions for the upper closures and lower closures on $\struct {S, \preccurlyeq}$ of an element $a$ of $S$.

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

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

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

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

Similarly for the closure operators on $\struct {S, \preccurlyeq}$ of a subset $T$ of $S$:

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

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

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

- $T^\succ := \bigcup \set {t^\succ: t \in T}$: 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.

The $\mathsf{Pr} \infty \mathsf{fWiki}$ style can be found in 2014: Nik Weaver: *Forcing for Mathematicians*.

It is a relatively recent innovation whose elegance and simplicity are compelling.

### 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

Yet other notations can be seen, for example:

- $\map {L_\prec} a$ for $a^\prec$
- $\map {L_\preccurlyeq} a$ for $a^\preccurlyeq$