Category:Definitions/Rays (Order Theory)

This category contains definitions related to rays in the context of Order Theory.
Related results can be found in Category:Rays (Order Theory).

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 $S$.

Open Ray

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

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

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

Closed Ray

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

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

$\set {x \in S: x \preccurlyeq a}$ (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 $a^\succ:= \set {x \in S: a \prec x}$

a closed ray $a^\succcurlyeq: \set {x \in S: a \preccurlyeq x}$

Downward-Pointing Ray

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

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

a closed ray $a^\preccurlyeq : \set {x \in S: x \preccurlyeq a}$