# Definition:Lower Closure/Element

## Definition

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 $a$.

## 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 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 $\left({S, \preccurlyeq}\right)$ is a well-ordered set, 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 on a general ordered set.

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