Does two isomorphic lattices always have the same Hasse diagram?

Question

I have done few examples of proving that lattices are isomorphic. I just draw their Hasse diagrams and if they are the same I say its isomorphic. Is there any case where two isomorphic latices don't have the same Hasse diagram?

Answer 1

Two lattices (or, more generally, posets) are isomorphic if and only if their order-relation is 'the same'. That is, if $(P, \le_P)$ and $(Q, \le_Q)$ are posets and $f : P \to Q$ is an isomorphism then $a \le_P b$ if and only if $f(a) \le_Q f(b)$. Thus they have the same Hasse diagram.