792 research outputs found
Weakly unambiguous morphisms
Weakly unambiguous morphism
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
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
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
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.....
Unambiguous morphic images of strings
Motivated by the research on pattern languages, we study a fundamental combinatorial question on morphisms in free semigroups: With regard to any string α over some alphabet we ask for the existence
of a morphism σ such that σ(α) is unambiguous, i.e. there is no morphism ρ with ρ ≠ σ and ρ(α) = σ(α). Our main result shows that a rich and natural class of strings is provided with unambiguous morphic images
Strict \infty-groupoids are Grothendieck \infty-groupoids
We show that there exists a canonical functor from the category of strict
\infty-groupoids to the category of Grothendieck \infty-groupoids and that this
functor is fully faithful. As a main ingredient, we prove that free strict
\infty-groupoids on a globular pasting scheme are weakly contractible.Comment: 22 pages, v2: revised according to referee's comments, in particular:
new organization of the pape
Unambiguous morphic images of strings
We study a fundamental combinatorial problem on morphisms in free semigroups: With
regard to any string α over some alphabet we ask for the existence of a morphism σ such
that σ(α) is unambiguous, i.e. there is no morphism T with T(i) ≠ σ(i) for some symbol
i in α and, nevertheless, T(α) = σ(α). As a consequence of its elementary nature, this
question shows a variety of connections to those topics in discrete mathematics which
are based on finite strings and morphisms such as pattern languages, equality sets and,
thus, the Post Correspondence Problem.
Our studies demonstrate that the existence of unambiguous morphic images essen-
tially depends on the structure of α: We introduce a partition of the set of all finite
strings into those that are decomposable (referred to as prolix) in a particular manner
and those that are indecomposable (called succinct). This partition, that is also known
to be of major importance for the research on pattern languages and on finite fixed
points of morphisms, allows to formulate our main result according to which a string α
can be mapped by an injective morphism onto an unambiguous image if and only if α is
succinct
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