In this talk I will give an overview on a way of defining classes of word relations called Synchronized relations. This framework yields an infinite number of classes living within the well-studied class of Rational relations, among which we can find the classes Automatic, Recognizable, or length-preserving relations.
A natural approach to defining binary word relations over a finite alphabet A is through two-tape finite state automata, which can be seen as regular languages L over the alphabet {1,2} x A, where (i,a) is interpreted as reading letter a from tape i. Thus, a word w of the language L denotes the pair (u_1,u_2) in A* x A* in which u_i is the projection of w onto i-labelled letters. While this formalism defines the well-studied class of Rational relations due to Nivat's Theorem (a.k.a. non-deterministic finite state transducers), enforcing restrictions on the reading regime from the tapes, that we call synchronization, yields various sub-classes of relations. Such synchronization restrictions are imposed through regular properties on the projection of the language onto {1,2}. In this way, for each regular language C contained in {1,2}*, one obtains a class Rel(C) of relations. I will show some recent results on this collection of classes of relations, based on joint work with MarĂa Emilia Descotte, Santiago Figueira, Leonid Libkin, and Gabriele Puppis.