# Meet Semilattice is Semilattice

## Theorem

Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.

Then $\struct {S, \wedge}$ is a semilattice.

## Proof

By definition of meet semilattice, $\wedge$ is closed.

The other three defining properties for a semilattice follow respectively from:

Meet is Commutative
Meet is Associative
Meet is Idempotent

Hence $\struct {S, \wedge}$ is a semilattice.

$\blacksquare$