This paper contains derivation of an asymptotic formula for the width (maximum size of an unordered subset) of a partial order that is the direct product of n orders each of size less than some bound c, and each having a nonempty set of ordered pairs. The elements of each factor can be represented as real numbers so that ordered pairs differ by at least one. If one gives each element an equal weight and associates with each factor the minimal variance such representation of it, the product of these orders may be corresponded with the sum of these random variables.
This sum obeys a central limit theorem, which gives the asymptotic formula: the width of the product order is asymptotic to its size divided by \(\sqrt(2\pi\) times the sum of the factor variances). Moreover for h of the form o(\(\sqrt{n})\), the union of h unordered sets is at most asymptotic to h times this number. That this is a lower bound follows easily by fleshing out the preceding description. That it is a lower bound is proven here by a difficult argument based upon the fact that for sufficiently large n most of the factors must occur very often for bounded c.