Definition:Ray (Order Theory)/Downward-Pointing

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

Let $\prec$ be the reflexive reduction of $\preccurlyeq$.

Let $a \in S$ be any point in $S$.

A downward-pointing ray is a ray which is bounded above:

an open ray $a^\prec := \left\{{x \in S: x \prec a}\right\}$
a closed ray $a^\preccurlyeq : \left\{{x \in S: x \preccurlyeq a}\right\}$

Also denoted as

The notations:

$\left({\gets \,.\,.\, a}\right)$ for $a^\prec$
$\left({\gets \,.\,.\, a}\right]$ for $a^\preccurlyeq$

can also be used.

Also see

  • Results about rays in the context of order theory can be found here.