Does there exist a lattice with a unique coatom, but whose unique coatom is not the second-from-top element?

Question

Does there exist a (necessarily infinite) lattice $L$ which has a unique coatom, but such that the unique coatom is not the second-from-top element? By second-from-top, I mean, if the top element was deleted from the lattice, the second-from-top would then be the new top element in the resulting partial order.

Answer 1

Start with the real interval $[0,1]$ under the usual order, and add a new element $c$ that is incomparable with any real number $x\in (0,1)$, but which is required to satisfy $0<c<1$. The element $c$ will be a unique coatom, but not be second-from-top.