Ordered sets

Definition 1(Definition).

Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following two properties

• Trichotomy: If $x\in S$ and $y\in S$, then one and only one of the following is true: $x<y,x=y,x>y$.
• Transitivity: If $x,y,z\in S$, if $x<y$ and $y<z$, then $x<z$.

Definition 2(Definition).

An ordered set is a set $S$ on which an order is defined.