# Definition:Lower Closure/Element

## Definition

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 $a$.

### Class Theory

In the context of class theory, the definition follows the same lines:

Let $A$ be a class under an ordering $\preccurlyeq$.

Let $a \in A$.

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

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

## Also known as

The **lower closure** of an element $a$ is also known as:

- the
**down-set**of $a$ - the
**down set**of $a$ - the
**lower set**of $a$ - the
**lower section**of $a$ - the
**set of preceding elements**to $a$

The terms **weak lower closure** and **weak down-set** are also encountered, so as explicitly to distinguish this from the strict lower closure of $a$.

When $\preccurlyeq$ is a well-ordering, the term **weak initial segment** is often used, and defined as a separate concept in its own right.

The notations $S_a$ or $\bar S_a$ are frequently then seen.

Some authors use the term **(weak) initial segment** to refer to the **lower closure** with respect to a general 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 $a^\preccurlyeq$

## Also see

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 14$: Order