# Definition:Strict Lower Closure/Element

## Definition

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

Let $a \in S$.

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

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

or:

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

That is, $a^\prec$ is the set of all elements of $S$ that strictly 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 strict lower closure of $a$ (in $A$) is defined as:

$a^\prec := \set {b \in A: b \preccurlyeq a \land a \ne b}$

or:

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

## Also known as

The strict lower closure of an element $a$ also goes by the names:

strict down-set
strict down set
initial segment (particularly when $\preccurlyeq$ is a well-ordering)
strict initial segment
set of (strictly) preceding elements to $a$

The term (strict) initial segment is usually seen in discussion of the properties of ordinals.

In this context, the notation $S_a$ or $\map s a$ can often be found for $a \in S$.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the term an initial segment of $S$ is specifically reserved for the strict lower closure of some element $a$ of $S$ under a well-ordering.

In particular, see Initial Segment of Natural Numbers.

## 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 $a^\preccurlyeq$