Dual space of a lattice as the completion of a Pervin space

The original motivation of this paper, as presented in [15], was to compute the dual space of a lattice of subsets of some free monoid A. According to StonePriestley duality, the dual space of a lattice can be identified with the set of its prime filters, but it is not always the simplest way to describe it. Consider for instance the Boolean algebra generated by the sets of the form uA, where u is a word. Its dual space is equal to the completion of A for the prefix metric and it can be easily identified with the set of finite or infinite words on A, a more intuitive description than prime filters. Elaborating on this idea, one may wonder whether the dual space of a given lattice of subsets of a space can always be viewed as a completion of some sort. The answer to this question is positive and known for a long time: for Boolean algebras, the solution is detailed as an exercise in Bourbaki [7, Exercise 12, p. 211]. In the lattice case, the appropriate setting for this question is a very special type of spaces, the so-called Pervin spaces, which form the topic of this paper. A Pervin space is a set X equipped with a set of subsets, called the blocks of the Pervin space. Blocks are closed under finite intersections and finite unions and hence form a lattice of subsets of X. Pervin spaces are thus easier to define than topological spaces or (quasi)-uniform spaces. As a consequence, most of the standard topological notions, like convergence and cluster points, specialisation order, filters and Cauchy filters, complete spaces and completion are much easier to define for Pervin spaces. The second motivation of this paper, also stemming from language theory, is the characterisation of classes of languages by inequations, which is briefly