Meet Absorbs Join

Theorem

Let $\left({S, \vee, \preceq}\right)$ be a join semilattice.

Let $\wedge$ denote meet.

Then $\wedge$ absorbs $\vee$.

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

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

Proof

By Dual Pairs (Order Theory), we observe that the theorem statement is dual to that of Join Absorbs Meet.

The result follows by the Global Duality Principle.

$\blacksquare$

Duality

The dual to this theorem is Join Absorbs Meet.