Maximal Element is Dual to Minimal Element

Theorem

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

Let $T \subseteq S$, and $a \in T$.

The following are dual statements:

$a$ is a maximal element of $T$

$a$ is a minimal element of $T$

Proof

By definition, $a$ is a maximal element of $T$ if and only if:

$\forall t \in T: a \preceq t$ implies $a = t$

The dual of this statement is:

$\forall t \in T: t \preceq a$ implies $a = t$

by Dual Pairs (Order Theory).

By definition, this means $a$ is a minimal element of $T$.

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

$\blacksquare$

Also see

• Duality Principle (Order Theory)