Prove that Idempotent property of lattices follows from commutative, associative and absorption property.

Question

I tried but I am not able to prove this. I am able to prove $a+a=a$ but not $a\cdot a=a$.

This is what i did so far:

$$a+(a\cdot b)=a+a=a.$$

Thus $a+a=a$.

Answer 1

You’ve already shown that $a+a=a$. and you know that $a\cdot(a+b)=a$ for all $b$; substitute $b=a$ into that absorption law and simplify using your previous result.