Absorption law

Statement

Let $L$ be a lattice with join $\vee$ and meet $\wedge$. Then $a \vee (a \wedge b) = a$ and $a \wedge (a \vee b) = a$ for all $a, b \in L$.

Remarks

In a lattice defined as a partially ordered set, the proof is short. Since $a \wedge b \le a$, the join of $a$ and $a \wedge b$ is $a$. The second law is dual. In the algebraic definition of a lattice, the two absorption laws are axioms, together with associativity and commutativity. The absorption laws imply idempotence: $a \vee a = a \vee (a \wedge (a \vee a)) = a$.

Explanations

L1 human

Absorption says that combining $a$ with something made from $a$ gives back $a$. For sets: $A \cup (A \cap B) = A$, because $A \cap B$ is already inside $A$.