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:

Closure
Associativity
Commutativity
Idempotence


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 $\vee$:

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

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

$\Box$


Commutativity

Let $x, y \in S$.

By definition of $\vee$:

\(\displaystyle x \vee y\) \(=\) \(\displaystyle \sup \set {x, y}\)
\(\displaystyle \) \(=\) \(\displaystyle \sup \set {y, x}\) Definition of Supremum of Set
\(\displaystyle \) \(=\) \(\displaystyle y \vee x\)

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

$\Box$


Idempotence

For all $x \in S$:

\(\displaystyle x \vee x\) \(=\) \(\displaystyle \sup \set {x, x}\)
\(\displaystyle \) \(=\) \(\displaystyle x\)

Hence $\vee$ is idempotent.

$\Box$


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

$\blacksquare$


Sources