Definition:Ray (Order Theory)
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This page is about Ray in the context of Order Theory. For other uses, see Ray.
Definition
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 $a^\preccurlyeq : \set {x \in S: x \preccurlyeq a}$
Also known as
A ray (either open or closed is also sometimes referred to as a half-line (either open or closed).
The Wirth interval notation is another notation used widely on $\mathsf{Pr} \infty \mathsf{fWiki}$:
- $\openint a \to$ for $a^\succ$
- $\openint \gets a$ for $a^\prec$
- $\hointr a \to$ for $a^\succcurlyeq$
- $\hointr \gets a$ for $a^\preccurlyeq$
can also be used.
Also see
- Definition:Order Topology: a topology whose sub-basis consists of open rays.
- Results about rays in the context of order theory can be found here.