Meet is Commutative

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

Join is Commutative

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Sources

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.23 \ \text {(a)}$