Meet is Associative

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Theorem

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


Then $\wedge$ is associative.


Proof

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

Then:

\(\displaystyle a \wedge \left({b \wedge c}\right)\) \(=\) \(\displaystyle \inf \left\{ {a, b \wedge c} \right\}\) Definition of meet
\(\displaystyle \) \(=\) \(\displaystyle \inf \left\{ {\inf \left\{ {a} \right\}, \inf \left\{ {b, c} \right\} } \right\}\) Infimum of Singleton
\(\displaystyle \) \(=\) \(\displaystyle \inf \left\{ {a, b, c} \right\}\) Infimum of Infima
\(\displaystyle \) \(=\) \(\displaystyle \inf \left\{ {\inf \left\{ {a, b} \right\}, \inf \left\{ {c} \right\} } \right\}\) Infimum of Infima
\(\displaystyle \) \(=\) \(\displaystyle \inf \left\{ {a, b} \right\} \wedge c\) Infimum of Singleton
\(\displaystyle \) \(=\) \(\displaystyle \left({a \wedge b}\right) \wedge c\) Definition of meet

Hence the result.

$\blacksquare$


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