Unambiguous Injective Morphisms in Free Groups

Reidenbach, Daniel; Day, Joel D.

Abstract

A morphism g is ambiguous with respect to a word u if there exists a second morphism h 6= g such that g(u) = h(u). Otherwise g is unambiguous with respect to u. Thus unambiguous morphisms are those for which the structure of the morphism is preserved in the image. Ambiguity has so far been studied for morphisms of free monoids, where several characterisations exist for the set of words u permitting an (injective) unambiguous morphism. In the present paper, we consider ambiguity of morphisms of free groups, and consider possible analogies to the existing characterisations in the free monoid. While a direct generalisation results in a trivial situation where all morphisms are ambiguous, we discuss some natural and well-motivated reformulations, and provide a characterisation of words in a free group that
permit a morphism which is “as unambiguous as possible”.

Citation

Reidenbach, D., & Day, J. D. (2022). Unambiguous Injective Morphisms in Free Groups. Information and Computation, 289(Part A), https://doi.org/10.1016/j.ic.2022.104946

• Journal Article Type: Article
• Acceptance Date: Jul 20, 2022
• Publication Date: Nov 22, 2022
• Journal: Information and Computation
• Print ISSN: 0890-5401
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 289
• Issue: Part A
• DOI: https://doi.org/10.1016/j.ic.2022.104946
• Keywords: Free groups; Automorphisms; Ambiguity of morphisms
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0890540122001018