We prove the ordered abelian group of monomials of the field of logarithmic exponential transseries is isomorphic to the additive reduct of the field itself and describe such an isomorphism (omega map); then discuss its relation with Conway's omega map on surreal numbers. In the process, a way of defining a "bounded version" of Hahn fields, starting from some additionl "bound" data, is described, and a decomposition theorem for the group of monomials of transseries is proven.