Let $S$ be a set. An ordering relation is a relation $\leq$ on $S$ such that, for every $a,b,c \in S$

• Either $a \leq b$ or $b \leq a$
• If $a \leq b$ and $b \leq c$ then $a \leq c$
• If $a \leq b$ and $b \leq a$ then $a = b$

Equivalently, an ordering relation is a relation $\leq$ on $S$ which makes the pair $(S,\leq)$ into a totally ordered set. Warning: In some cases, an author may use the term ``ordering relation'' to mean a partial order instead of a total order.

Given an ordering relation $\leq$ one can define a relation $<$ by: $a < b$ if $a \leq b$ and $a \neq b$ The opposite ordering is the relation $\geq$ given by: $a \geq b$ if $b \leq a$ and the relation $>$ is defined analogously.