Float is different because nobody really expect exact answers from Float, and also because errors accumulate in complex way.
I still wouldn't want Float to belong to Group. If some domains claim to fulfill the axioms the Group imposes, then they actually should. No exception.
For TaylorSeries some results are exact, and it is easier to control approximation error.
It's still not a Ring. We should rather build a better category system where we take computability more into account. We can only model the computable part of mathematics correctly. We should understand that we will never have a 1-to-1 correspondence between mathematical concepts and concepts in FriCAS. Neither domains nor categories. Why do we have so many polynomial domains? Because it makes sense in FriCAS. Mathematics doesn't care much about complexity issues.
Expression is different, because various folks have quite different idea what Expression should really be (as I wrote, I think we should have a few expression domains with different semantics and with new names).
Oh, yes, I am very much in favour of your suggestion. In particular, clarifying the semantics is quite an important part. But, actually, we don't really need it. If one wants clear semantics, then FriCAS has a clear way to construct an appropriate algebra for a given task. The problem is that users just give an expression and expect the system to figure out the nasty details. Expression domains are much more difficult to handle than any complicated algebra that can be build with FriCAS domain constructors.
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