Let A be a set with a strict partial order ≺; let x ∈ A. Suppose that we wish to find a ma...

Let A be a set with a strict partial order ≺; let x ∈ A. Suppose that we wish to find a maximal simply ordered subset B of A that contains x. One plausible way of attempting to define B is to let B equal the set of all those elements of A that are comparable with x;

But this will not always work. In which of Examples 1 and 2 will this procedure succeed and in which will it not?