  The notion of a difference hierarchy, first introduced by Hausdorff, plays an
important role in many areas of mathematics, logic and theoretical computer
science such as descriptive set theory, complexity theory, and the theory of
regular languages and automata. From a lattice theoretic point of view, the
difference hierarchy over a bounded distributive lattice stratifies the Boolean
algebra generated by it according to the minimum length of difference chains
required to describe the Boolean elements. While each Boolean element is given
by a finite difference chain, there is no canonical such writing in general. We
show that, relative to the filter completion, or equivalently, the lattice of
closed upsets of the dual Priestley space, each Boolean element over the
lattice has a canonical minimum length decomposition into a Hausdorff
difference. As a corollary each Boolean element over a (co-)Heyting algebra has
a canonical difference chain. With a further generalization of this result
involving a directed family of adjunctions with meet-semilattices, we give an
elementary proof of the fact that a regular language is given by a Boolean
combination of purely universal sentences using arbitrary numerical predicates
if and only if it is given by a Boolean combination of purely universal
sentences using only regular numerical predicates.
