Meet Semilattice has Greatest Element iff has Identity

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Theorem

Let $\struct{S, \wedge, \preceq}$ be a meet semilattice.

Let $s \in S$.

Then:

$s$ is the greatest element of $S$

if and only if

$s$ is the identity in $\struct{S, \wedge}$.

Proof

This is the dual statement of Join Semilattice has Smallest Element iff has Identity by Dual Pairs (Order Theory).

The result follows from the Duality Principle.

$\blacksquare$