Join Semilattice is Semilattice

Theorem
Let $\struct {S, \vee, \preceq}$ be a join semilattice.

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

Proof
Recall the definition of join semilattice:

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 $\struct {S, \vee}$ is a semilattice.

Also see

 * Meet Semilattice is Semilattice