Greatest Element is Dual to Smallest Element

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

Let $a \in S$.

The following are dual statements:


 * $a$ is the greatest element of $S$
 * $a$ is the smallest element of $S$

Proof
By definition, $a$ is the greatest element of $S$ :


 * $\forall b \in S: b \preceq a$

The dual of this statement is:


 * $\forall b \in S: a \preceq b$

by Dual Pairs (Order Theory).

By definition, this means $a$ is the smallest element of $S$.

The converse follows from Dual of Dual Statement (Order Theory).

Also see

 * Duality Principle (Order Theory)