contradiction in the behavior of WordMorphism

[videlec]

@seblabbe @farbulu94

Sage considers Words("ab") as different from Words("ba"). As a consequence the following word morphism is not an endomorphism

sage: s = WordMorphism('a->ba,b->a', domain=Words('ab'), codomain=Words('ba'))
sage: s.is_endomorphism()
False

In particular, sage complains about it in method such as

sage: s.is_primitive()
Traceback (most recent call last):
...
TypeError: self (=a->ab, b->a) is not an endomorphism

However, it does not complain when it comes to composition which looks like a bug to me since it breaks functoriality of incidence matrix

sage: s * s
WordMorphism: a->abb, b->b
sage: (s * s).incidence_matrix() == s.incidence_matrix() * s.incidence_matrix()
False

[Rahul22031982]

Sage distinguishes between Words('ab') and Words('ba'), so even if they contain the same letters, the order matters. This causes WordMorphism('a->ba,b->a', domain=Words('ab'), codomain=Words('ba')) to not be an endomorphism, which leads to errors in methods like .is_primitive() and breaks expected matrix behavior in composition.

[Rahul22031982]

To fix this, simply set both domain and codomain to have exactly the same alphabet in the same order:
s = WordMorphism('a->ba, b->a', domain=Words('ab'), codomain=Words('ab'))