Glas List: Re: [glas] ring_traits and vector space

From: Doug Gregor (dgregor_at_[hidden])
Date: 2005-01-11 10:31:06

On Jan 11, 2005, at 1:49 AM, Toon Knapen wrote:

> Doug Gregor wrote:
>> In the context of a ring, however, 0.0 is the
>> identity and 1.0 is the unity and there is no confusion. This may be 
>> best expressed by calling the above structure "ring_traits" instead 
>> of "value_traits".
>
> A ring has also multiplicative identity (in addition to be being an 
> abelian group (R,+)) so 'identity' is still confusing.

The ambiguity is resolved in some textbooks by calling the 
multiplicative identity the "unity", and the additive identity the 
"identity".

> But related to the question if the value_type's should be in a ring is 
> whether the vectors and matrices (in the glas library) should be 
> defined in a vector-space or not. If so, this requires the 
> value_type's to be in a field (and thus not restricted to a ring).

This seems desirable to me.

> OTOH we can also see vectors and matrices as 1D and 2D containers. 
> This would allow for instance to use a matrix to store e.g. 
> function-pointers in a 2D structure or have std::pair as value_type 
> for vectors. The requirements on the operations on the value_type then 
> would only depend on the _operations_ one performs on the vector or 
> matrix (e.g. the value_type should have a operator- if you want to 
> substract 2 vectors).

It's not really an either-or situation. We can say that the vectors and 
matrices of the glas library model the Vector Space concept when the 
value_type is a field (the common case), or some weaker concept when 
the value_type  is not a field.

        Doug