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On compositions of root-to-frontier tree transformations

It is well known that the family of (nondeterministic) root-to-frontier tree transformations is not closed with respect to the composition, see [2]. In this paper we introduce the notion of ^-synchronized root-to-frontier tree transducer. Transducers of this type are capable of inducing all the relations which are compositions of k root-to-frontier tree transformations. Conversely, we shall show that any relation induced by a ^-synchronized tree transducer is a composition of k root-to-frontier tree transformations. We mention that similar results are obtained by M. Dauchet in his dissertation [1] using the theory of magmoids.