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.In their more general form, our equations are of the form u → v, where u and v are profinite words. The first result not only subsumes Eilenberg-Reiterman's theory of varieties and their subsequent extensions, but it shows for instance that any class of regular languages defined by a fragment of logic closed under conjunctions and disjunctions (first order, monadic second order, temporal, etc.) admits an equational description. In particular, the celebrated McNaughtonSchützenberger characterisation of first order definable languages by the aperiodicity condition x ω = x ω+1 , far from being an isolated statement, now appears as an elegant instance of a very general result.How is this equational theory related to duality? The connection between profinite words and Stone spaces was already discovered by Almeida [2], [3, Theorem 3.6.1], but Pippenger [14] was the first to formulate it in terms of Stone duality. Almeida (implicitely) and Pippenger (explicitely) both observed that the Boolean algebra of regular languages over A * is dual to the Stone space A * , the set of profinite words. Pippenger actually came very close to our first result, since he mentioned that this duality extends to a one-to-one correspondence between Boolean algebras of regular languages and quotients of A * . Our first result is the full-fledged consequence of the similar one-to-one correspondence for all lattices of languages provided by Priestley duality.However, this link to duality theory is in fact much stronger and encompasses not only the underlying lattices and spaces involved but also the algebraic operations including the product of profinite words. That is the content of our

We consider the four families of recognizable, synchronous, deterministic rational and rational subsets of a direct product of free monoids. They form a strict hierarchy and we investigate the following decision problem: given a relation in one of the families, does it belong to a smaller family? We settle the problem entirely when all monoids have a unique generator and fill some gaps in the general case. In particular, adapting a proof of Stearns, we show that it is recursively decidable whether or not a deterministic subset of an arbitrary number of free monoids is recognizable. Also we exhibit a single exponential algorithm for determining if a synchronous relation is recognizable