# Upper Semilattice on Classical Set is Semilattice

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## Theorem

Let $\struct {S, \vee}$ be an upper semilattice on a classical set $S$.

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

## Proof

To show that the algebraic structure $\struct {S, \vee}$ is a semilattice, the following need to be verified:

In order:

### Closure

By definition of an upper semilattice:

- $\forall x, y \in S: \sup \set {x, y} \in S$

Since $x \vee y = \sup \set {x, y} \in S$ for all $x, y \in S$:

- $\struct {S, \vee}$ is closed.

$\Box$

### Associativity

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

By definition of $\vee$:

\(\ds \paren {x \vee y} \vee z\) | \(=\) | \(\ds \sup \set {x, y} \vee z\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sup \set {\sup \set {x, y}, z}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sup \set {x, y, z}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sup \set {x, \sup \set {y, z} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds x \vee \sup \set {y, z}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds x \vee \paren {y \vee z}\) |

Hence $\struct {S, \vee}$ is associative.

$\Box$

### Commutativity

Let $x, y \in S$.

By definition of $\vee$:

\(\ds x \vee y\) | \(=\) | \(\ds \sup \set {x, y}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sup \set {y, x}\) | Definition of Supremum of Set | |||||||||||

\(\ds \) | \(=\) | \(\ds y \vee x\) |

Hence $\struct {S, \vee}$ is commutative.

$\Box$

### Idempotence

For all $x \in S$:

\(\ds x \vee x\) | \(=\) | \(\ds \sup \set {x, x}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds x\) |

Hence $\vee$ is idempotent.

$\Box$

Having explicitly verified all prerequisites, it follows that $\struct {S, \vee}$ is a semilattice.

$\blacksquare$

## Sources

- 1981: Stanley Burris and H.P. Sankappanavar:
*A Course in Universal Algebra*: $\text {II} \ \S 1$ Example $(7)$