# Meet is Idempotent

## Theorem

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

Then $\wedge$ is idempotent.

## Proof

Let $a \in S$ be arbitrary.

Then:

 $\displaystyle a \wedge a$ $=$ $\displaystyle \inf \set {a, a}$ Definition of Meet (Order Theory) $\displaystyle$ $=$ $\displaystyle \inf \set a$ Axiom of Extension $\displaystyle$ $=$ $\displaystyle a$ Infimum of Singleton

Hence the result.

$\blacksquare$