This paper presents a new result in the equational theory of regular languages, which emerged from lively discussions between the authors about Stone and Priestley duality. Let us call lattice of languages a class of regular languages closed under finite intersection and finite union. The main results of this paper (Theorems 5.2 and 6.1) can be summarized in a nutshell as follows: A set of regular languages is a lattice of languages if and only if it can be defined by a set of profinite equations. The product on profinite words is the dual of the residuation operations on regular languages.
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