Join is Commutative

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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