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

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Sources

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