Definition:Lower Closure

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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.


Also see