On Tuesday, June 12, 2018 at 7:24:45 AM UTC-7, Jeroen Demeyer wrote:
>
> I'd like to say that the following should be coercions:
> R -> R[x,y]
> R[x] -> R[x,y]
> R[y] -> R[x,y]
> R[x][y] -> R[x,y]
>
> But not the following:
> R[x,y] -> R[x][y]
> R[x,y] -> R[y,x]
> R[x,y][z] -> R[x][y,z]
>

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That looks reasonable to me. Some of the maps in the second list might be
conversions?
I agree with John that it should at least be possible to create all those
maps for sure, and hopefully easily. Possibly a nice test for seeing if the
list of coercions and conversions present is sufficient is to see if all
those maps can be defined by giving a list of images of generators.
Another point to keep in mind:
sage: A=PolynomialRing(QQ,["x","y"],order="deglex")
sage: B=PolynomialRing(QQ,["x","y"],order="lex")
are different rings in sage. What coercions and conversions exist between
them?
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