Join is Idempotent

Theorem

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

Then $\vee$ is idempotent.

Proof

Let $a \in S$ be arbitrary.

Then:

\(\ds a \vee a\)

\(=\)

\(\ds \sup \set {a, a}\)

Definition of Join

\(\ds \)

\(=\)

\(\ds \sup \set a\)

Definition of Set

\(\ds \)

\(=\)

\(\ds a\)

Supremum of Singleton

Hence the result.

$\blacksquare$

Also see

• Meet is Idempotent

This article is complete as far as it goes, but it could do with expansion. In particular: Needs to be phrased using a "partial operation" to remove the condition that $\vee$ is a total operation You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.23 \ \text {(a)}$