# Upper Semilattice on Classical Set is Semilattice It has been suggested that this page or section be merged into Join Semilattice is Semilattice. (Discuss)

## 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.

$\Box$

### Associativity

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

By definition of $\vee$:

 $\displaystyle \left({x \vee y}\right) \vee z$ $=$ $\displaystyle \sup \left\{ {x, y}\right\} \vee z$ $\displaystyle$ $=$ $\displaystyle \sup \left\{ {\sup \left\{ {x, y}\right\}, z}\right\}$ $\displaystyle$ $=$ $\displaystyle \sup \left\{ {x, y, z}\right\}$ $\displaystyle$ $=$ $\displaystyle \sup \left\{ {x, \sup \left\{ {y, z}\right\} }\right\}$ $\displaystyle$ $=$ $\displaystyle x \vee \sup \left\{ {y, z}\right\}$ $\displaystyle$ $=$ $\displaystyle x \vee \left({y \vee z}\right)$

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

$\Box$

### Commutativity

Let $x, y \in S$.

By definition of $\vee$:

 $\displaystyle x \vee y$ $=$ $\displaystyle \sup \left\{ {x, y}\right\}$ $\displaystyle$ $=$ $\displaystyle \sup \left\{ {y, x}\right\}$ Definition of Supremum $\displaystyle$ $=$ $\displaystyle y \vee x$

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

$\Box$

### Idempotence

For all $x \in S$:

 $\displaystyle x \vee x$ $=$ $\displaystyle \sup \left\{ {x, x}\right\}$ $\displaystyle$ $=$ $\displaystyle x$

Hence $\vee$ is idempotent.

$\Box$

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

$\blacksquare$