#11506: Fix the infinity ring.
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Reporter: vbraun | Owner: AlexGhitza
Type: defect | Status: needs_info
Priority: blocker | Milestone: sage-6.3
Component: algebra | Resolution:
Keywords: | Merged in:
Authors: Volker Braun | Reviewers: Peter Bruin
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/vbraun/infinity_ring | 3531287276d95f0a60b762c4dc5475bee4860cba
Dependencies: #13125 | Stopgaps:
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Comment (by tscrim):
Replying to [comment:38 pbruin]:
> `RIF` is definitely not a field (and neither are `RR` and `RDF`). They
are Sage objects that approximate the field '''R''' of real numbers in
different ways, but none of these Sage objects satisfies the axioms of a
field.
Since there is `oo` in `RIF`, I agree with you, it's actually not a field.
Although I would say that if we remove `oo`, I would say it is a field.
But this is a separate point, because the break is in the addition, not
the multiplication. So would you say `RIF` is an additive abelian group?
I also found some strange behavior with `RIF`:
{{{
sage: oo in RIF
True
sage: RIF(oo)
[+infinity .. +infinity]
sage: RIF(oo) / RIF(oo)
[.. NaN ..]
sage: RIF(oo) / RIF(0)
[-infinity .. +infinity]
sage: RIF(oo) * RIF(0)
0
}}}
> I don't understand this. Addition of `+Infinity` and `-Infinity` is
undefined in the infinity ring, or is this what you mean by "less
structure"?
Yes (another example would be the tropical semiring).
> No, the operation `4 // 2` is defined in `ZZ`, but not after coercion to
`Zmod(8)`, for example.
In this case, division is not a part of the structure of `ZZ`, so perhaps
I should amend my statement above to when you can ''always'' do operation
`#` (i.e. it is guaranteed by the category). For `RIF`, subtraction is
guaranteed to work.
--
Ticket URL: <http://trac.sagemath.org/ticket/11506#comment:39>
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