Ordering

An ordering is a special kind of relation, other than the concept of a relation we require three properties before we can define it.

Required properties

These are restated from the relation page for convenience

Symmetric

Note: this isn't actually needed, it is stated to allow the reader to contrast with antisymmetric and asymmetric

A relation $R$ in $A$ is symmetric if for all $a,b\in A$ we have that $aRb\implies bRa$ - a property of equivalence relations

Antisymmetric

A binary relation $R$ in $A$ is antisymmetric if for all $a,b\in A$ we have $$aRb\text{ and }bRA\implies a=b$$
Symmetric implies elements are related to each other, antisymmetric implies only the same things are related to each other.

Reflexive

For a relation $R$ and for all $a\in A$ we have $aRa$ - $a$ is related to itself.

Transitive

A relation $R$ in $A$ is transitive if for all $a,b,c\in A$ we have $$[aRb\text{ and }bRc\implies aRc]$$

Asymmetric

A relation $S$ in $A$ is asymmetric if $aSb\implies(b,a)\notin S$, for example $<$ has this property, we can have $a<b$ or $b<a$ but not both.

Definition

If we have "just an ordering" it refers to a partial ordering.

Partial Ordering

A binary relation that is antisymmetric, reflexive and transitive is a partial ordering. An example of a partial ordering is $\le$, notice $a\le b$ and $$b\le a\implies a=b$$

Strict ordering

A relation $S$ in $A$ is a strict ordering if it is asymmetric and transitive.