Hi Joseph,

For what it's forth, I may tell what is done in SageManifolds (metaticket 
#18528 <https://trac.sagemath.org/ticket/18528>): SageManifolds involves 
many parent classes. All of them are implemented as direct subclasses of 
Parent. Their algebraic structures (group, free module, non-free module, 
commutative algebra) are set by the category (i.e. via the keyword argument 
'category' in Parent.__init__).
The corresponding element classes are implemented as subclasses of Sage 
specific classes: MultiplicativeGroupElement, ModuleElement, 
CommutativeAlgebraElement.

Best wishes,

Eric.

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