Meet is Associative

Theorem

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

Then $\wedge$ is associative.

Proof

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

Then:

$\ds a \wedge \paren {b \wedge c}$

$=$

$\ds \inf \set {a, b \wedge c}$

Definition of Meet

$\ds $

$=$

$\ds \inf \set {\inf \set a, \inf \set {b, c} }$

Infimum of Singleton

$\ds $

$=$

$\ds \inf \set {a, b, c}$

Infimum of Infima

$\ds $

$=$

$\ds \inf \set {\inf \set {a, b}, \inf \set c}$

Infimum of Infima

$\ds $

$=$

$\ds \inf \set {a, b} \wedge c$

Infimum of Singleton

$\ds $

$=$

$\ds \paren {a \wedge b} \wedge c$

Definition of Meet

Hence the result.

$\blacksquare$

Also see

Join is Associative

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
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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