Unambiguous Injective Morphisms in Free Groups

Reidenbach, Daniel; Day, Joel D.

doi:10.1016/j.ic.2022.104946

Unambiguous Injective Morphisms in Free Groups

Reidenbach, Daniel; Day, Joel D.

Authors

Daniel Reidenbach d.reidenbach@keele.ac.uk

Joel D. Day

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”.

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