Covering and separation for logical fragments with modular predicates

For every class $mathscr{C}$ of word languages, one may associate a decision problem called $mathscr{C}$-separation. Given two regular languages, it asks whether there exists a third language in $mathscr{C}$ containing the first language, while being disjoint from the second one. Usually, finding an algorithm deciding $mathscr{C}$-separation yields a deep insight on $mathscr{C}$. We consider classes defined by fragments of first-order logic. Given such a fragment, one may often build a larger class by adding more predicates to its signature. In the paper, we investigate the operation of enriching signatures with modular predicates. Our main theorem is a generic transfer result for this construction. Informally, we show that when a logical fragment is equipped with a signature containing the successor predicate, separation for the stronger logic enriched with modular predicates reduces to separation for the original logic. This result actually applies to a more general decision problem, called the covering problem.