>
> In the same vein, shouldn't base_ring be removed from Parent as well ?
>
> sage: Set([1,2,3]).base_ring()
>
> The return value is None, which is arguably correct, for a generic set has 
> no base ring, but it might be surprising when using tab completion to 
> discover functionalities to see that a base_ring() method exists for such 
> objects.
>

+1 to that. Many other methods of Parent should be moved somewhere else 
(is_exact, inject_variables,gens_dict*

Moreover, many method names shoulf be changed (made longer, more explicit): 
while the meaning of Parent.a_method can be clear, it is at times very 
difficult, when listing the methods of a more specific object, to 
understand what the same method does. Because then it is not obvious at all 
that a_method refers to the object 'understood as a Parent'. When you have 
a poset or a group in yours hands, your think of it as a group, and not 
necessarily as "a parent".

Nathann

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