Maximal Element need not be Unique

Theorem
Let $\struct {S, \preccurlyeq}$ be an ordered set.

It is possible for $S$ to have more than one maximal element.

Proof
Consider the set $T$ defined as:
 * $T = \set {0, 1}$

Let $S$ be defined as:
 * $S := \powerset T \setminus T$

where $\powerset T$ denotes the power set of $T$.

That is:
 * $S = \set {\O, \set 0, \set 1}$

Let $\preccurlyeq$ be the relation defined on $S$ as:
 * $\forall a, b \in S: a \preccurlyeq b \iff a \subseteq b$

That is, $\preccurlyeq$ is the subset relation on $S$.

From Subset Relation is Ordering, $\struct {S, \preccurlyeq}$ is an ordered set.

Let $a \in S$ such that $\set 1 \preccurlyeq a$.

Then by inspection it is apparent that:
 * $a = \set 1$

That is, $\set 1$ is a maximal element of $\struct {S, \preccurlyeq}$.

Similarly, let $a \in S$ such that $\set 0 \preccurlyeq a$.

Then by inspection it is apparent that:
 * $a = \set 0$

That is, $\set 0$ is also a maximal element of $\struct {S, \preccurlyeq}$.

Hence $S$ has more than one maximal element.

Also see

 * Smallest Element is Unique