On the decomposition of k-valued rational relations

We give a new, and hopefully more easily understandable, structural proof of the decomposition of a $k$-valued transducer into $k$ unambiguous functional ones, a result established by A. Weber in 1996. Our construction is based on a lexicographic ordering of computations of automata and on two coverings that can be build by means of this ordering. The complexity of the construction, measured as the number of states of the transducers involved in the decomposition, improves the original one by one exponential. Moreover, this method allows further generalisation that solves the problem of decomposition of rational relations with bounded length-degree, which was left open in Weber's paper.