Definition:Ray (Order Theory)/Closed

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Definition

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

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


The following sets are called closed rays or closed half-lines:

$\left\{{x \in S: a \preccurlyeq x}\right\}$ (the upper closure of $a$), denoted $a^\succcurlyeq$
$\left\{{x \in S: x \preccurlyeq a}\right\}$ (the lower closure of $a$), denoted $a^\preccurlyeq$.


Also known as

A closed ray is also sometimes referred to as a closed half-line.

The notations:

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

can also be used.


Also see

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