#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:40 pbruin]:
> As Volker pointed out, it isn't, even if we remove infinity. And since
it isn't an additive Abelian group, it certainly isn't a field either.
Moreover, the set of non-zero, non-infinite elements is not a group under
multiplication. For example, the interval [-1, 1] has neither an additive
nor a multiplicative inverse.
> ...
> That depends on what you mean by "work":
> {{{
> sage: RIF(oo) - RIF(oo)
> [.. NaN ..]
> }}}
> Maybe this should return `[-infinity .. +infinity]`, then it would
cooperate nicely with the future support for "intervals in the infinity
ring" (comments 29 and 30 above).
From my understanding, elements in RIF aren't intervals as sets, they are
a single point that's guaranteed to be within in the interval. Thus if we
fix a real element `x` in an interval, say `[-1, 1]`, then `x - x == 0`
should be true (which I would say that it does not is currently a bug).
However doing something like `RIF(-1,1) - RIF(-1, 1)` gets two
(essentially) random numbers in the interval `[-1, 1]` and subtracts them
and `RIF` tells us the result is guaranteed to lie in `[-2, 2]`.
Well...at least if we throw out the exact values of `+/-oo` and the
`NaN`...(and it's essentially a field to the average user)
We also have this behavior with the current branch:
{{{
sage: InfinityRing(RIF(oo) - RIF(oo))
A positive finite number
sage: InfinityRing(float("NaN"))
A negative finite number
}}}
At least it maps this over. So to add to the question list, how do we
handle this too? Keep it as is?
> I still don't get your point. The tropical semiring is a semiring, while
InfinityRing is not a semiring.
Okay, I'm going to take a deep breath and try to make I'm not missing any
cases or over simplifying things.
In `RIF` (with `[.. NaN ..]`), we have an additive abelian semigroup as we
have an associative binary operation.
For `InfinityRing`, we have 5 elements: `[-oo, n, 0, p, oo]` where `n` and
`p` are finite negative and positive numbers respectively. The following
additions are defined (up to commutation):
{{{
-oo + [n, 0, p] -> -oo
n + [n, 0] -> n
p + [0, p] -> p
oo + [n, 0, p] -> oo
}}}
but none of the others (and multiplication is worse). So it doesn't have
any fully defined binary operations on the entire set.
I didn't miss any cases, right? If so, then this is when I say "less
structure".
--
Ticket URL: <http://trac.sagemath.org/ticket/11506#comment:41>
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