Maximal Element of Chain is Greatest Element

Theorem

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

Let $C$ be a chain in $S$.

Let $m$ be a maximal element of $C$.

Then $m$ is the greatest element of $C$.

Proof

Let $x \in C$.

Since $m$ is maximal in $C$, $m \nprec x$.

Since $C$ is a chain, $x = m$ or $x \prec m$.

Thus for each $x \in C$, $x \preceq m$.

Therefore $m$ is the greatest element of $C$.

$\blacksquare$

Also see

• Minimal Element of Chain is Smallest Element

Sources

• 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S\text I.3$