Definition:Strict Lower Closure/Set
Definition
Let $\struct {S, \preceq}$ be an ordered set or a preordered set.
Let $T \subseteq S$.
The strict lower closure of $T$ (in $S$) is defined as:
- $\ds T^\prec := \bigcup \set {t^\prec: t \in T}$
where $t^\prec$ denotes the strict lower closure of $t$ in $S$.
That is:
- $T^\prec := \set {u \in S: \exists t \in T: u \prec t}$
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
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