Meet is Commutative

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice.


Then $\wedge$ is commutative.


Proof

Let $a, b \in S$ be arbitrary.

Then:

\(\displaystyle a \wedge b\) \(=\) \(\displaystyle \inf \left\{ {a, b}\right\}\) Definition of Meet
\(\displaystyle \) \(=\) \(\displaystyle \inf \left\{ {b, a}\right\}\) Axiom of Extension
\(\displaystyle \) \(=\) \(\displaystyle b \wedge a\) Definition of Meet

Hence the result.

$\blacksquare$


Also see