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Bounds are given on the number of broken derived terms (a variant of Antimirov's 'partial derivatives') of a rational expression E. It is shown that this number is less than or equal to 2ℓ(E) + 1 in the general case, where ℓ(E) is the literal length of the expression E, and that the classical bound ℓ(E) + 1 which holds for partial derivatives also holds for… (More)

It is known that any rational abstract numeration system is faithfully, and effectively, represented by an N-rational series. A simple proof of this result is given which yields a representation of this series which in turn allows a simple computation of the value of words in this system and easy constructions for the recognition of recognisable sets of… (More)

We define and study here the class of rational functions that are finite union of sequential functions. These functions can be realized by cascades of sequential transducers. After showing that cascades of any height are equivalent to cascades of height at most two and that this class strictly contains sequential functions and is strictly contained in the… (More)

We prove that the function that maps a word of a rational language onto its successor for the radix order in this language is a finite union of co-sequential functions.

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