I read about Matrix Algebra over Tropical Semiring and found that there is
"tropical determinant" for this purpose (Page 3 : Eq 4
<http://www.math.uni-konstanz.de/~michalek/may22.pdf>). This formulation
does not use (additive) inversion.
I was also trying different operations between tropical elements and other
elements(like Integers). It looks like there is a need for implementation
of coercion maps between few datatypes and tropical elements.
One can check this by running few commands like
sage: T = TropicalSemiring(QQ)
sage: a = T(1); b = T(0)
sage: a + 0; b + 0
sage: a * 0; b * 0
sage: a + 1; b + 1
sage: a * 1; b * 1
Coercion maps and models are crucial for polynomial element implementation.
For example : between
'Symbolic Ring' and 'Tropical semiring over Rational Field'
'Univariate Polynomial Ring in x over Tropical semiring over Rational
Field' and 'Tropical semiring over Rational Field'
'Tropical semiring over Rational Field' and 'Integer Ring'
On Friday, March 29, 2024 at 12:16:44 AM UTC+5:30 tcscrims wrote:
> 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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