Proposition 3

Let E be an ordered set and let E$'$ be the disjoint union of E and a set $\{a\}$ consisting of a single element. Then there exists a unique ordering on E$'$ which induces the given ordering on E and for which $a$ is the greatest element of E$'$.

For if G is the graph of the ordering on E, the graph of an ordering on E$'$ which satisfies the conditions of the Proposition must be the union G$'$ of G and the set of all pairs $(x, a)$ where $x \in E'$; conversely, it is clear that G$'$ is the graph of an ordering on E$'$ which satisfies the given conditions.

¶ The ordered set E$'$ is said to be obtained by adjoining a greatest element $a$ to E (cf. Exercise 3).

¶ A subset $A$ of a preordered set $E$ is said to be cofinal (resp. coinitial) in $E$ if for every $x\in E$ there exists $y\in A$ such that $x\leq y$ (resp. $y\leq x$). To say that an ordered set $E$ has a greatest (resp. least) element therefore means that $E$ has a cofinal (resp. coinitial) subset consisting of a single element.