An order relation (also called an order or ordering) is a binary relation $~$\le$~$ on a set $~$S$~$ that can be used to order the elements in that set.
An order relation satisfies the following properties:
- For all $~$a \in S$~$, $~$a \le a$~$. (the reflexive property)
- For all $~$a, b \in S$~$, if $~$a \le b$~$ and $~$b \le a$~$, then $~$a = b$~$. (the antisymmetric property)
- For all $~$a, b, c \in S$~$, if $~$a \le b$~$ and $~$b \le c$~$, then $~$a \le c$~$. (the transitive property)
A set that has an order relation is called a partially ordered set (or "poset"), and $~$\le$~$ is its partial order.
Totality of an order
There is also a fourth property that distinguishes between two different types of orders:
- For all $~$a, b \in S$~$, either $~$a \le b$~$ or $~$b \le a$~$ or both. (the [total_relation total] property)
The total property implies the reflexive property, by setting $~$a = b$~$.
If the order relation satisfies the total property, then $~$S$~$ is called a Totally ordered set, and $~$\le$~$ is its total order.
Well-ordering
A fifth property that extends the idea of a "total order" is that of the well-ordering:
- For every subset $~$X$~$ of $~$S$~$, $~$X$~$ has a least element: an element $~$x$~$ such that for all $~$y \in X$~$, we have $~$x \leq y$~$.
Well-orderings are very useful: they are the orderings we can perform induction over. (For more on this viewpoint, see the page on [structural_induction].)
Derived relations
The order relation immediately affords several other relations.
Reverse order
We can define a reverse order $~$\ge$~$ as follows: $~$a \ge b$~$ when $~$b \le a$~$.
Strict order
From any poset $~$(S, \le)$~$, we can derive a strict order $~$<$~$, which disallows equality. For $~$a, b \in S$~$, $~$a < b$~$ when $~$a \le b$~$ and $~$a \neq b$~$. This strict order is still antisymmetric and transitive, but it is no longer reflexive.
We can then also define a reverse strict order $~$>$~$ as follows: $~$a > b$~$ when $~$b \le a$~$ and $~$a \neq b$~$.
Incomparability
In a poset that is not totally ordered, there exist elements $~$a$~$ and $~$b$~$ where the order relation is undefined. If neither $~$a \leq b$~$ nor $~$b \leq a$~$ then we say that $~$a$~$ and $~$b$~$ are incomparable, and write $~$a \parallel b$~$.
Cover relation
From any poset $~$(S, \leq)$~$, we can derive an underlying cover relation $~$\prec$~$, defined such that for $~$a, b \in S$~$, $~$a \prec b$~$ whenever the following two conditions are satisfied:
- $~$a < b$~$.
- For all $~$s \in S$~$, $~$a \leq s < b$~$ implies that $~$a = s$~$.
Simply put, $~$a \prec b$~$ means that $~$b$~$ is the smallest element of $~$S$~$ which is strictly greater than $~$a$~$. $~$a \prec b$~$ is pronounced "$~$a$~$ is covered by $~$b$~$", or "$~$b$~$ covers $~$a$~$", and $~$b$~$ is said to be a cover of $~$a$~$.
Comments
Dylan Hendrickson
Any relation satisfying 1-3 is a partial order, and the corresponding set is a poset. A total order is a special kind of partial order defined by also satisfying 4.
Joe Zeng
So effectively all order relations are partial order relations?