The Many Faces of a Translation

First-order translations have recently been characterized as the maps computed by aperiodic single-valued nondeterministic finite transducers (NFTs). It is shown here that this characterization lifts to “V-translations” and “V-single-valued-NFTs”, where V is an arbitrary monoid pseudovariety that is closed under reversal. More strikingly, two-way V-transducer are introduced, and the following three models are shown exactly equivalent to Eilenberg’s classical notion of a bimachine when V is a group variety or when V is the variety of aperiodic monoids: V-translations, V-single-valued-NFTs and two-way V-transducers.