Intervals (pairs of fuzzy values), triples, etc.: can we thus get an arbitrary ordering?

Traditional fuzzy logic uses real numbers as truth values. This description is not always adequate, so in interval-valued fuzzy logic, we use pairs (t^-, t⁺) of real numbers, t^- ⩽ t⁺, to describe a truth value. To make this description even more adequate, instead of using real numbers to described each value t^- and t⁺, we can use intervals, and thus get fuzzy values which can be described by 4 real numbers each. We can iterate this procedure again and again. The question is: can we get an arbitrary partially ordered set in this manner or an arbitrary lattice? In this paper, we show that although we cannot thus generate arbitrary lattices, we can actually generate an arbitrary partially ordered set in this manner. In this sense, the “intervalization” operation is indeed universal