The unambiguity of segmented morphisms

Abstract

This paper studies the ambiguity of morphisms in free monoids. A morphism σ is said to be ambiguous with respect to a string α if there exists a morphism τ which differs from σ for a symbol occurring in α, but nevertheless satisfies τ (α) = σ(α); if there is no such τ then σ is called unambiguous. Motivated by the recent initial paper on the ambiguity of morphisms, we introduce the definition of a so-called segmented morphism σ 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 (σ n) of those finite strings over an infinite alphabet with respect to which σ n is unambiguous, and we comprehensively describe its relation to any U (σ m), m = n. 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.

Unambiguous 1-uniform morphisms

“…Hence, intuitively speaking, unambiguous morphisms have a desirable, namely structure-preserving, property in such a context, and therefore previous literature on the ambiguity of morphisms mainly studies the question of the existence of unambiguous morphisms for arbitary words. In the initial paper, Freydenberger, Reidenbach and Schneider [6] show that there exists an unambiguous nonerasing morphism with respect to a word α if and only if α is not a fixed point of a nontrivial morphism, i. e., there is no morphism φ satisfying φ(α) = α and, for a symbol x in α, φ(x) = x. Freydenberger and Reidenbach [5] study those sets of words with respect to which so-called segmented morphisms are unambiguous, and these results lead to a refinement of the techniques used in [6]. Schneider [15] and Reidenbach and Schneider [14] investigate the existence of unambiguous erasing morphisms -i. e., morphisms that may map symbols to the empty word.…”

“…Note that[6,5] also deal with unambiguous nonerasing morphisms, but they use a stronger notion of unambiguity that is based on arbitrary monoid morphisms. Hence, they call a morphism σ unambiguous only if there is no other -erasing or nonerasingmorphism τ satisfying τ (α) = σ(α).…”

Unambiguous 1-Uniform Morphisms

“…Hence, intuitively speaking, unambiguous morphisms have a desirable, namely structure-preserving, property in such a context, and therefore previous literature on the ambiguity of morphisms mainly studies the question of the existence of unambiguous morphisms for arbitary words. In the initial paper, Freydenberger, Reidenbach and Schneider [5] show that there exists an unambiguous nonerasing morphism with respect to a word α if and only if α is not a fixed point of a nontrivial morphism, i. e., there is no morphism φ satisfying φ (α) = α and, for a symbol x in α, φ (x) = x. Freydenberger and Reidenbach [4] study those sets of words with respect to which socalled segmented morphisms are unambiguous, and these results lead to a refinement of the techniques used in [5]. Schneider [13] and Reidenbach and Schneider [12] investigate the existence of unambiguous erasing morphisms -i. e., morphisms that may map symbols to the empty word.…”

“…Note that[5,4] also deal with unambiguous nonerasing morphisms, but they use a stronger notion of unambiguity that is based on arbitrary monoid morphisms. Hence, they call a morphism σ unambiguous only if there is no other -erasing or nonerasing -morphism τ satisfying τ(α) = σ (α).…”