# Join is Commutative

Jump to navigation
Jump to search

## Theorem

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

Then $\vee$ is commutative.

## Proof

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

Then:

\(\displaystyle a \vee b\) | \(=\) | \(\displaystyle \sup \left\{ {a, b}\right\}\) | Definition of join | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sup \left\{ {b, a}\right\}\) | Axiom of Extension | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle b \vee a\) | Definition of join |

Hence the result.

$\blacksquare$

## Also see