Definition:Ray (Order Theory)

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This page is about upper closure or lower closure in a totally ordered set. For other uses, see Definition:Ray.

Definition

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 $S$.


Open Ray

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

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


Closed Ray

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 $a^\preccurlyeq$.


Upward-Pointing Ray

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 $a^\succcurlyeq: \left\{{x \in S: a \preccurlyeq x}\right\}$


Downward-Pointing Ray

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 known as

A ray (either open or closed is also sometimes referred to as a half-line (either open or closed).

The notations:

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