Variétés et fonctions rationnelles

AbstractWe say that a rational (resp. a subsequential) function α 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 α−1(L) divides that of L, for each rational language L on the outputs. In the second case, α−1(L) is a group-language for each group language L; equivalently, α 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