Following claim is incorrect: It can be shown that any nonempty family R of linear extensions is a realizer if and only if, for every critical pair (x,y) of P, y <i x for some order <i in R.
If we take 2 copies of , each with the default ordering, but no relation between the copies, then there are obviously no critical pairs at all. Now assume that we extend this to a total ordering by requiring that all the elements of the first copy are smaller than all the elements of the second copy. The claim says that this single linearly ordered set is a realizer of our original partial order, which is clearly untrue. Finiteness (of P) seems to be sufficient, but not necessary.