Join Absorbs Meet

Theorem
Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice.

Let $\vee$ denote join.

Then $\vee$ absorbs $\wedge$.

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


 * $a \vee \left({a \wedge b}\right) = a$

Proof
By Ordering in terms of Join, we have that:


 * $a \vee \left({a \wedge b}\right) = a$ iff $a \wedge b \preceq a$

The result thus follows from Meet Precedes Operands.

Duality
The dual to this theorem is Meet Absorbs Join.