Join is Commutative

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Theorem

Let $\struct {S, \vee, \preceq}$ be a join semilattice.


Then $\vee$ is commutative.


Proof

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

Then:

\(\ds a \vee b\) \(=\) \(\ds \sup \set {a, b}\) Definition of Join (Order Theory)
\(\ds \) \(=\) \(\ds \sup \set {b, a}\) Definition of Set Equality
\(\ds \) \(=\) \(\ds b \vee a\) Definition of Join (Order Theory)

Hence the result.

$\blacksquare$


Also see


This article is complete as far as it goes, but it could do with expansion.
In particular: Needs to be phrased using a "partial operation" to remove the condition that $\vee$ is a total operation
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