Meet is Commutative

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Theorem

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


Then $\wedge$ is commutative.


Proof

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

Then:

\(\ds a \wedge b\) \(=\) \(\ds \inf \set {a, b}\) Definition of Meet
\(\ds \) \(=\) \(\ds \inf \set {b, a}\) Definition of Set Equality
\(\ds \) \(=\) \(\ds b \wedge a\) Definition of Meet

Hence the result.

$\blacksquare$


Also see


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 $\wedge$ is a total operation
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