Definition:Ray (Order Theory)/Upward-Pointing

From ProofWiki
Jump to navigation Jump to search


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 $S$.

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

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

Also denoted as

The notations:

$\left({a \,.\,.\, \to}\right)$ for $a^\succ$
$\left[{a \,.\,.\, \to}\right)$ for $a^\succcurlyeq$

can also be used.

Also see

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