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.

Also see

 * Join Semilattice is Semilattice