Meet is Idempotent

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.


Then $\wedge$ is idempotent.


Proof

Let $a \in S$ be arbitrary.

Then:

\(\displaystyle a \wedge a\) \(=\) \(\displaystyle \inf \set {a, a}\) Definition of Meet (Order Theory)
\(\displaystyle \) \(=\) \(\displaystyle \inf \set a\) Axiom of Extension
\(\displaystyle \) \(=\) \(\displaystyle a\) Infimum of Singleton

Hence the result.

$\blacksquare$


Also see