Join is Associative

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Theorem

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


Then $\vee$ is associative.


Proof

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

Then:

\(\displaystyle a \vee \paren {b \vee c}\) \(=\) \(\displaystyle a \vee \sup \set {b, c}\) Definition of Join (Order Theory)
\(\displaystyle \) \(=\) \(\displaystyle \sup \set {\sup \set a, \sup \set {b, c} }\) Supremum of Singleton
\(\displaystyle \) \(=\) \(\displaystyle \sup \set {a, b, c}\) Supremum of Suprema
\(\displaystyle \) \(=\) \(\displaystyle \sup \set {\sup \set {a, b}, \sup \set c}\) Supremum of Suprema
\(\displaystyle \) \(=\) \(\displaystyle \sup \set {a, b} \vee c\) Supremum of Singleton
\(\displaystyle \) \(=\) \(\displaystyle \paren {a \vee b} \vee c\) Definition of Join (Order Theory)

Hence the result.

$\blacksquare$


Also see