Join is Idempotent

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Then $\vee$ is idempotent.


Proof

Let $a \in S$ be arbitrary.

Then:

\(\displaystyle a \vee a\) \(=\) \(\displaystyle \sup \set {a, a}\) Definition of Join
\(\displaystyle \) \(=\) \(\displaystyle \sup \set a\) Axiom of Extension
\(\displaystyle \) \(=\) \(\displaystyle a\) Supremum of Singleton

Hence the result.

$\blacksquare$


Also see