Join is Associative

Theorem

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

Then $\vee$ is associative.

Proof

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

Then:

$\ds a \vee \paren {b \vee c}$

$=$

$\ds a \vee \sup \set {b, c}$

Definition of Join (Order Theory)

$\ds $

$=$

$\ds \sup \set {\sup \set a, \sup \set {b, c} }$

Supremum of Singleton

$\ds $

$=$

$\ds \sup \set {a, b, c}$

Supremum of Suprema

$\ds $

$=$

$\ds \sup \set {\sup \set {a, b}, \sup \set c}$

Supremum of Suprema

$\ds $

$=$

$\ds \sup \set {a, b} \vee c$

Supremum of Singleton

$\ds $

$=$

$\ds \paren {a \vee b} \vee c$

Definition of Join (Order Theory)

Hence the result.

$\blacksquare$

Also see

Meet is Associative

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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Sources

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.23 \ \text {(a)}$