Are all lattices chains?

Question

Are all lattices chains? I think that is true because a chain is a poset where we can compare any two elements. A lattice is a poset where every subset has a lub and a gld. So, by reducing the size of the subset to two, having a lub and glb essentially means they are comparable and hence a chain. Is my intuition wrong?

Answer 1

You are using the same type of argument as in the following "proof".

All points of the plane are aligned. Indeed, two points are aligned. Take now $n+1$ points $a_0, \ldots, a_n$. By the induction hypothesis, the $n$ points $a_0, \ldots, a_{n-1}$ are aligned and so are $a_1, \ldots, a_n$. Since there is only one line containing $a_1, a_2$, the $n+1$ points are aligned.

Can you see the gap in this argument?