The unambiguity of segmented morphisms

• Authors: Dominik D. Freydenberger;Daniel Reidenbach
• Venue: Discrete Applied Mathematics
• Year: 2009

This paper studies the ambiguity of morphisms in free monoids. A morphism @s is said to be ambiguous with respect to a string @a if there exists a morphism @t which differs from @s for a symbol occurring in @a, but nevertheless satisfies @t(@a)=@s(@a); if there is no such @t then @s is called unambiguous. Motivated by the recent initial paper on the ambiguity of morphisms, we introduce the definition of a so-called segmented morphism @s"n, which, for any n@?N, maps every symbol in an infinite alphabet onto a word that consists of n distinct factors in ab^+a, where a and b are different letters. For every n, we consider the set U(@s"n) of those finite strings over an infinite alphabet with respect to which @s"n is unambiguous, and we comprehensively describe its relation to any U(@s"m), mn. Thus, our work features the first approach to a characterisation of sets of strings with respect to which certain fixed morphisms are unambiguous, and it leads to fairly counter-intuitive insights into the relations between such sets. Furthermore, it shows that, among the widely used homogeneous morphisms, most segmented morphisms are optimal in terms of being unambiguous for a preferably large set of strings. Finally, our paper yields several major improvements of crucial techniques previously used for research on the ambiguity of morphisms.