Join Absorbs Meet

Theorem

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

Let $\vee$ denote join.

Then $\vee$ absorbs $\wedge$.

That is, for all $a, b \in S$:

$a \vee \paren {a \wedge b} = a$

Proof

By Ordering in terms of Join, we have that:

$a \vee \paren {a \wedge b} = a$ if and only if $a \wedge b \preceq a$

The result thus follows from Meet Precedes Operands.

$\blacksquare$

Duality

The dual to this theorem is Meet Absorbs Join.

Sources

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