# Difference hierarchies and duality with an application to formal languages

## TL;DR

This paper explores the structure of difference hierarchies in lattices and their applications to formal languages, providing canonical decompositions and new insights into regular language representations.

## Contribution

It introduces a canonical minimum length decomposition of Boolean elements in lattices relative to filter completion and extends this to applications in formal language theory.

## Key findings

- Canonical difference chain decompositions exist for Boolean elements over lattices.
- Boolean elements over (co-)Heyting algebras have canonical difference chains.
- Regular languages can be characterized by Boolean combinations of universal sentences with specific predicates.

## Abstract

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.