# Definition:Strict Lower Closure

(Redirected from Definition:Strict Lower Set)

## Definition

### Strict Lower Closure of Element

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

Let $a \in S$.

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

$a^\prec := \left\{{b \in S: b \preccurlyeq a \land a \ne b}\right\}$

or:

$a^\prec := \left\{{b \in S: b \prec a}\right\}$

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

### Strict Lower Closure of Subset

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

Let $T \subseteq S$.

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

$T^\prec := \displaystyle \bigcup \left\{{t^\prec: t \in T}\right\}$

where $t^\prec$ denotes the strict lower closure of $t$ in $S$.

That is:

$T^\prec := \left\{ {u \in S: \exists t \in T: u \prec t}\right\}$

## 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 $S$.

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

$\displaystyle T^\preccurlyeq := \bigcup \set {t^\preccurlyeq: t \in T}$: the lower closure of $T \in S$: everything in $S$ that precedes some element of $T$
$\displaystyle T^\succcurlyeq := \bigcup \set {t^\succcurlyeq: t \in T}$: the upper closure of $T \in S$: everything in $S$ that succeeds some element of $T$
$\displaystyle T^\prec := \bigcup \set {t^\prec: t \in T}=$: the strict lower closure of $T \in S$: everything in $S$ that strictly precedes some element of $T$
$\displaystyle T^\succ := \bigcup \set {t^\succ: t \in T}$: 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.

### 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

The strict lower closure also goes by the names:

• Strict down-set
• Strict down set
• Strict lower set
• Initial segment (in the context of a well-ordered set)