Upper Semilattice on Classical Set is Semilattice

Theorem
Let $\left({S, \vee}\right)$ be an upper semilattice on a classical set $S$.

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

Proof
To show that the algebraic structure $\left({S, \vee}\right)$ is a semilattice, the following need to be verified:


 * Closure
 * Associativity
 * Commutativity
 * Idempotence

In order:

Closure
By definition of an upper semilattice:


 * $\forall x, y \in S: \sup \left\{{x, y}\right\} \in S$

Since $x \vee y = \sup \left\{{x, y}\right\} \in S$ for all $x, y \in S$:
 * $\left({S, \vee}\right)$ is closed.

Associativity
Let $x, y, z \in S$.

By definition of $\vee$:

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

Commutativity
Let $x, y \in S$.

By definition of $\vee$:

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

Idempotence
For all $x \in S$:

Hence $\vee$ is idempotent.

Having explicitly verified all prerequisites, it follows that $\left({S, \vee}\right)$ is a semilattice.