Join Semilattice is Semilattice

Theorem
Let $\left({S, \vee, \preceq}\right)$ be a join semilattice.

Then $\left({S, \vee}\right)$ is a semilattice.

Proof
By definition of join semilattice, $\vee$ is closed.

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


 * Join is Commutative
 * Join is Associative
 * Join is Idempotent

Hence $\left({S, \vee}\right)$ is a semilattice.

Also see

 * Meet Semilattice is Semilattice