Radix Cross-Sections for Length Morphisms

We prove that the radix cross-section of a rational set for a length morphism, and more generally for a rational function from a free monoid into N, is rational, a property that does not hold any more if the image of the function is a subset of a free monoid with two or more generators. proceedings short version1 The purpose of this paper is to give a positive answer to a problem left open in an old paper by the second author ([11]) and to prove the following property, a refinement of the Cross-Section Theorem ([3]): Proposition 1. The radix cross-section of a rational set for a length morphism is rational. By ‘rational set’ we mean rational set of a free monoid A∗ and by ‘length morphism’, a morphism from A∗ into N, or, which is the same, into {x}∗, the one generator free monoid. Let us take for instance the alphabet A = {a, b} , the morphism θ : A∗ → {x}∗ defined by aθ = x2 and bθ = x3 and the rational set R = (ba∗)∗ . The lexicographic cross-section of R for θ is 1 + ba∗(1 + b) , the set of words in R such that each one is the smallest in the lexicographic order in its class modulo the map equivalence of θ (this smallest element exists, even if the lexicographic order is not a well-ordering, since every class is finite). The radix cross-section of R for θ is 1 + b (1 + a+ a2)b∗ , the set of words in R obtained if we replace the lexicographic order by the radix order (sometimes called shortlex or length-lexicographic order), which is a well-ordering. That the lexicographic cross-section of a rational set is rational follows from results that are recalled in the next section. That the radix cross-section of a rational set is rational is thus established in this paper. As for the Cross-Section Theorem, Proposition 1 has a dual, and equivalent, formulation which is better suited for both proof and generalisation. ∗LIGM, Université Paris-Est. †LTCI, CNRS / ENST, Paris. A longer version with full proofs is available on the web page of the authors.