Variétés et fonctions rationnelles

We say that a rational (resp. a subsequential) function tl from a free monoid into another one is in the variety of monoids V if it may realized by some unambiguous (resp. subsequential) transducer whose monoid of transitions is in V. We characterize these functions when V is the variety of aperiodic monoids, and the variety of groups. In the first case, the period of am ’ (15) divides that of L, for each rational language L on the outputs. In the second case, a-‘(L) is a group-language for each group language L; equivalently, a is continuous for the pro-finite topology. Examples of such functions are: the multiplication by a given number in a given basis, which is aperiodic; the division, which is a group-function.