The question is:


(i) Let $L$ be a distributive lattice and let $a,b,c\in L$. Prove that


($a\lor b$ = $c\lor b$ & $a\land b$ = $c\land b$) => $a$=$c$. $(*)$


(ii) Find elements $a,b,c$ in $M_3$ violating $(*)$. Do the same for $N_5$.


(iii) Deduce that a lattice $L$ is distributive if and only if $(*)$ holds $\forall$ $a,b,c\in L$.


My Attempt:


Well i managed to prove the first part just fine.


I am not sure about the second one, I was thinking of choosing $a=2$,$b=3$,$c=5$ for $M_3$.They do violate $(*)$, but I am not sure whether this is correct or not.
If it is correct then probably the same example should work for $N_5$ as well.


Also i am stuck with the converse of the third part,i.e., if $(*)$ is satified then show that $L$ is a distributive lattice.


I considered, $a\land (b\lor c)$ = $a\land (a\lor b)$ =$a$, how do I go further ?