# Meet is Commutative

## Theorem

Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice.

Then $\wedge$ is commutative.

## Proof

Let $a, b \in S$ be arbitrary.

Then:

 $\displaystyle a \wedge b$ $=$ $\displaystyle \inf \left\{ {a, b}\right\}$ Definition of Meet $\displaystyle$ $=$ $\displaystyle \inf \left\{ {b, a}\right\}$ Axiom of Extension $\displaystyle$ $=$ $\displaystyle b \wedge a$ Definition of Meet

Hence the result.

$\blacksquare$