A morphism g is ambiguous with respect to a word u if there exists a morphism h ≠ g such that g(u) = h(u). The ambiguity of morphisms has so far been studied in a free monoid. In the present paper, we consider the ambiguity of morphisms of the free group. Firstly, we note that a direct generalisation results in a trivial problem. We provide a natural reformulation of the problem along with a characterisation of elements of the free group which have an associated unambiguous injective morphism. This characterisation matches an existing result known for the free monoid. Secondly, we note a second formulation of the problem which leads to a non-trivial situation: when terminal symbols are permitted. In this context, we investigate the ambiguity of the morphism erasing all non-terminal symbols. We provide, for any alphabet, a pattern which can only be mapped to the empty word exactly by this morphism. We then generalize this construction to give, for any morphism g, a pattern α such that h(α) is the empty word if and only if h = g.
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