Meet is Idempotent

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Theorem

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


Then $\wedge$ is idempotent.


Proof

Let $a \in S$ be arbitrary.

Then:

\(\ds a \wedge a\) \(=\) \(\ds \inf \set {a, a}\) Definition of Meet (Order Theory)
\(\ds \) \(=\) \(\ds \inf \set a\) Definition of Set
\(\ds \) \(=\) \(\ds a\) Infimum of Singleton

Hence the result.

$\blacksquare$


Also see




Sources