The unambiguity of segmented morphisms

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 σn, 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(σn) of those finite strings over an infinite alphabet with respect to which σn is unambiguous, and we comprehensively describe its relation to any U(σm), 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

The unambiguity of segmented morphisms

The unambiguity of segmented morphism

Unambiguous 1-Uniform Morphisms

A morphism h is unambiguous with respect to a word w if there is no other morphism g that maps w to the same image as h. In the present paper we study the question of whether, for any given word, there exists an unambiguous 1-uniform morphism, i.e., a morphism that maps every letter in the word to an image of length 1.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Weakly Unambiguous Morphisms

A nonerasing morphism sigma is said to be weakly unambiguous with respect to a word w if sigma is the only nonerasing morphism that can map w to sigma(w), i.e., there does not exist any other nonerasing morphism tau satisfying tau(w) = sigma(w). In the present paper, we wish to characterise those words with respect to which there exists such a morphism. This question is nontrivial if we consider so-called length-increasing morphisms, which map a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions

Weakly unambiguous morphisms

Weakly unambiguous morphism

Unambiguous 1-uniform morphisms

A morphism σ is unambiguous with respect to a word α if there is no other morphism τ that maps α to the same image as σ. In the present paper we study the question of whether, for any given word, there exists an unambiguous 1-uniform morphism, i.e., a morphism that maps every letter in the word to an image of length 1

On restricting the ambiguity in morphic images of words

For alphabets Delta_1, Delta_2, a morphism g : Delta_1* to Delta_2* is ambiguous with respect to a word u in Delta_1* if there exists a second morphism h : Delta_1* to Delta_2* such that g(u) = h(u) and g not= h. Otherwise g is unambiguous. Hence unambiguous morphisms are those whose structure is fully preserved in their morphic images. A concept so far considered in the free monoid, the first part of this thesis considers natural extensions of ambiguity of morphisms to free groups. It is shown that, while the most straightforward generalization of ambiguity to a free monoid results in a trivial situation, that all morphisms are (always) ambiguous, there exist meaningful extensions of (un)ambiguity which are non-trivial - most notably the concepts of (un)ambiguity up to inner automorphism and up to automorphism. A characterization is given of words in a free group for which there exists an injective morphism which is unambiguous up to inner automorphism in terms of fixed points of morphisms, replicating an existing result for words in the free monoid. A conjecture is presented, which if correct, is sufficient to show an equivalent characterization for unambiguity up to automorphism. A rather counterintuitive statement is also established, that for some words, the only unambiguous (up to automorphism) morphisms are non-injective (or even periodic). The second part of the thesis addresses words for which all non-periodic morphisms are unambiguous. In the free monoid, these take the form of periodicity forcing words. It is shown using morphisms that there exist ratio-primitive periodicity forcing words over arbitrary alphabets, and furthermore that it is possible to establish large and varied classes in this way. It is observed that the set of periodicity forcing words is spanned by chains of words, where each word is a morphic image of its predecessor. It is shown that the chains terminate in exactly one direction, meaning not all periodicity forcing words may be reached as the (non-trivial) morphic image of another. Such words are called prime periodicity forcing words, and some alternative methods for finding them are given. The free-group equivalent to periodicity forcing words - a special class of C-test words - is also considered, as well as the ambiguity of terminal-preserving morphisms with respect to words containing terminal symbols, or constants. Moreover, some applications to pattern languages and group pattern languages are discussed

Weakly unambiguous morphisms

A nonerasing morphism σ is said to be weakly unambiguous with respect to a word s if σ is the only nonerasing morphism that can map s to σ(s), i. e., there does not exist any other nonerasing morphism τ satisfying τ(s) = σ(s). In the present paper, we wish to characterise those words with respect to which there exists such a morphism. This question is nontrivial if we consider so-called length-increasing morphisms, which map a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions

Conditions on the existence of unambiguous morphisms

A morphism α is (strongly) unambiguous with respect to a word α if there is no other morphism τ that maps α to the same image as σ. Moreover, α is said to be weakly unambiguous with respect to a word α if σ is the only nonerasing morphism.....