How to prove the absorption law in a lattice?

Question

It is easy to prove the absorption law in a lattice by taking particular example. Let $(A, \mid)$ be a lattice where $A=\{2, 4, 8\}$. Then $2\wedge(2\vee 4)=2\wedge\sup\{2, 4\}=2\wedge 4=\inf\{2, 4\}=2$. Similarly, $2\vee(2\wedge 4)=2$ holds. But instead of taking $2$ and $4$, if we take any pair of elements $a$ and $b$ in lattice $L$ then how do we prove that

• $a\wedge(a\vee b)=a$ and

• $a\vee(a\wedge b)=a$ ?

Answer 1

Assuming you're using the poset definition of lattice . . .

Let $c = a \land (a \lor b)$.

The goal is to show $c=a$.

From $c = a \land (a \lor b)$, it's immediate that $c \le a$.

But also, $a \le a$, and $a \le a\lor b$, hence $a \le c$.

Therefore $c = a$.

An analogous argument proves the other absorption law.

You try it.

Answer 2

$a\vee (a\wedge b) \\ = (a\wedge T) \vee (a \wedge b) \\= a \wedge (T \vee b) \\= a \wedge T \\= a$

Note : in the third line i used the inverse of distributive law