# Presenting de Groot duality of stably compact spaces

## TL;DR

This paper provides a constructive, point-free approach to de Groot duality in stably compact spaces using strong proximity lattices and introduces strong continuous entailment relations for analyzing duals.

## Contribution

It introduces strong continuous entailment relations as a new tool for representing and analyzing de Groot duals of stably compact spaces.

## Key findings

- De Groot duals can be identified via duals of presentations.
- Constructs powerlocales, patch topology, and valuation spaces.
- Approach simplifies reasoning about duality in stably compact spaces.

## Abstract

We give a constructive account of the de Groot duality of stably compact spaces in the setting of strong proximity lattice, a point-free representation of a stably compact space. To this end, we introduce a notion of strong continuous entailment relation, which can be thought of as a presentation of a strong proximity lattice by generators and relations. The new notion allows us to identify de Groot duals of stably compact spaces by analysing the duals of their presentations. We carry out a number of constructions on strong proximity lattices using strong continuous entailment relations and study their de Groot duals. The examples include various powerlocales, patch topology, and the space of valuations. These examples illustrate the simplicity of our approach by which we can reason about the de Groot duality of stably compact spaces.