One needs to be *very* careful about what operations mean. -T(2), the
(tropical) additive inverse of 2, is *not* T(-2). There is no way to have
min(2, a) = \infty (note that this is the (tropical) additive unit).
I am not sure how (or if) determinants over the tropical semiring are
defined, but one could define it by having the sign included in each term.
So for the 2x2 case, it could be T(a*b) + T(-b*d), but I don’t know if this
makes det a group homomorphism.
Best,
Travis
On Wednesday, March 27, 2024 at 2:17:39 AM UTC+9 anime...@iiits.in wrote:
> This problem can be seen clearly if we use *Matrix_generic_dense* as
> element class for matrix space.
>
>
> On Tuesday, March 26, 2024 at 9:42:45 PM UTC+5:30 Animesh Shree wrote:
>
>> I was going through this code and got error. But I could not understand
>> why this is the case.
>>
>>
>>
>>
>> sage: T = TropicalSemiring(QQ)
>> sage: T(1)
>> 1
>> sage: T(2)
>> 2
>> sage: T(-2)
>> -2
>> sage: -T(2)
>>
>> ---------------------------------------------------------------------------
>> ArithmeticError Traceback (most recent call
>> last)
>> Cell In[26], line 1
>> ----> 1 -T(Integer(2))
>>
>> File ~/sage/src/sage/rings/semirings/tropical_semiring.pyx:278, in
>> sage.rings.semirings.tropical_semiring.TropicalSemiringElement.__neg__()
>> 276 if self._val is None:
>> 277 return self
>> --> 278 raise ArithmeticError("cannot negate any non-infinite
>> element")
>> 279
>> 280 cpdef _mul_(left, right) noexcept:
>>
>> ArithmeticError: cannot negate any non-infinite element
>> sage:
>>
>>
>> It looks like starting with T(-2) and reaching to -2 from T(2) by
>> comparing with zero(+Inf) are different things.
>> T(-2) = -2
>> T(2) -T(2) = T.zero(+inf) = T(2) + (-T(2))
>>
>> My doubt is : if we cannot negate the elements, then how can we compute
>> the determinant of a Matrix over Tropical Semiring.
>> For example in 2x2 matrix : [[a,b], [c,d]] the determinant should be ad -
>> bc
>> But can it be expressed as ad + (-bc) -->> min ( add(a,d), -1*add(b,c)
>> )?
>>
>> In fact we connot even do matrix subtraction directly.
>> What can be done in these cases??
>> On Thursday, March 21, 2024 at 8:48:26 PM UTC+5:30 Animesh Shree wrote:
>>
>>> Hello Sir,
>>>
>>> I have submitted my proposal.
>>> Please review it and let me know necessary updates and improvements.
>>>
>>> I want to verify soundness of my approach and extend the proposal for
>>> Multivariate Polynomials.
>>>
>>> Thank You
>>> Animesh Shree
>>> On Tuesday, March 12, 2024 at 12:26:38 AM UTC+5:30 tcscrims wrote:
>>>
>>>> Mathematically speaking, you can always weaken axioms. However, there
>>>> are some extra advantages that additive groups have that commutative
>>>> semirings don't have (mainly 0, the additive identity).
>>>>
>>>> That being said, there isn't anything prevent you from constructing the
>>>> appropriate categories. It would be good to have a more specific use-case
>>>> in mind, but that isn't necessary. However, one should be careful with the
>>>> name because it would conflict with what "most" people would call an
>>>> algebra (which is why we have MagmaticAlgebras).
>>>>
>>>> Best,
>>>> Travis
>>>>
>>>>
>>>> On Tuesday, March 12, 2024 at 2:57:29 AM UTC+9 anime...@iiits.in wrote:
>>>>
>>>>> Hello Sir,
>>>>>
>>>>> I was going through "Algebras" and I had a doubt.
>>>>> Does MagmaticAlgebra and AssociativeAlgebra have to be implemented
>>>>> over Ring only?
>>>>> I went through internet and google uses rings to define those
>>>>> algebras, but the axioms that those algebra follow (Unital, Associative)
>>>>> are also preserved by commutative semirings.
>>>>> Would it be good to define MagmaticAlgebra and AssociativeAlgebra over
>>>>> other objects that follow those axioms too or come-up with alternative
>>>>> Algebra?
>>>>>
>>>>
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