Definition 9

Two elements $x$, $y$ of a preordered set E are said to be comparable if the relation “$x\leqslant y$ or $y\leqslant x$” is true. A set E is said to be totally ordered if it is ordered and if any two elements of E are comparable. The ordering on E is then said to be a total ordering, and the corresponding order relation a total order relation.

If $x$ and $y$ are elements of a totally ordered set, then $x=y$ or $x<y$ or $x>y$; the negation of $x\leqslant y$ is thus $x>y$.

An ordering on E is a total ordering if and only if its graph G satisfies the relation $G\cup\overset{-1}{G}=E\times E$, as well as the relations $G\circ G=G$ and $G\cap\overset{-1}{G}=\Delta$.

Examples