# Meet is Associative

## 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.

$\blacksquare$