First order formulas with modular predicates

Two results by Schützenberger (1965) and by McNaughton and Papert (1971) lead to a precise description of the expressive power of first order logic on words interpreted as ordered colored structures. In this paper, we study the expressive power of existential formulas and of Boolean combinations of existential formulas in a logic enriched by modular numerical predicates. We first give a combinatorial description of the corresponding regular languages, and then give an algebraic characterization in terms of their syntactic morphisms. It follows that one can effectively decide whether a given regular language is captured by one these two fragments of first order logic. The proofs rely on nontrivial techniques of semigroup theory: stamps, derived categories and wreath products