Definition:Ray (Order Theory)/Upward-Pointing

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

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

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

An upward-pointing ray is a ray which is bounded below:

an open ray $a^\succ:= \set {x \in S: a \prec x}$
a closed ray $a^\succcurlyeq: \set {x \in S: a \preccurlyeq x}$

Also denoted as

The notations:

$\openint a \to$ for $a^\succ$
$\hointr a \to$ for $a^\succcurlyeq$

can also be used.

Also see

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