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?
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