Definition:Closure under Ordering/Notation

Notation for Upper Closure and Lower Closure
Let $\left({S, \preccurlyeq}\right)$ be an ordered set or a preordered set.

On 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 $T$.

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.

The style can be found in.

It is a relatively recent innovation whose elegance and simplicity are compelling.