Glas List: Re: [glas] pure algebraic concepts cfr additive

From: Karl Meerbergen (Karl.Meerbergen_at_[hidden])
Date: 2005-05-27 01:34:48

Hi,

I am happy with Toon and Peter's contributions. It allows for some 
freedom in the creation of operators that lead to a group, e.g. element 
wise products of vectors and matrices (which can be implemented with a 
functor element_prod() or something of this kind).

Peter, have you had any chance to look at the richardson iteration example?

Have a nice day,

Karl

Peter Gottschling wrote:

>
> On May 26, 2005, at 3:22 PM, Toon Knapen wrote:
>
>> I'm currently wondering if it's a good idea to consider AdditiveMagma 
>> to be a refinement of the Magma concept. This might sound strange but 
>> let me explain.
>>
>> For the pure algebraic concepts, we always talk about a set S and 
>> some operation-type which is conform to a BinaryFunctor.
>> In the case of an AdditiveMagma, we don't need a seperate 
>> BinaryFunctor but instead we have an 'operator+(const T&)'. The 
>> operator+ is defined on the elements of the set.
>
>
> Of course, this looks strange in the first place that a single-type 
> concept refines a multi-type concept. By definition a refinement is an 
> addition of requirements, which implies that every type T that model 
> concept C2 necessarily models C1 if C2 is a refinement of C1. To have 
> this property between AdditiveMagma and Magma we need an extra tool, 
> the default functor. As I defined the concepts and the default 
> functors, every type that models AdditiveMagma is always a model of 
> Magma using the default functor.
>
> We can discuss if this is really a refinement in the pure sense of the 
> definition. We could may be call it semi-refinement or weak refinement 
> for distinction because it uses a tool. The important point is that it 
> works for implementations, i.e. every type that model some additive 
> concept can _always_ call a template function requiring the 
> corresponding pure algebraic function. And this is from my prospective 
> more important than whether we call it refinement or semi-refinement.
>
> BTW, we shall also check whether we can implement this (semi-) 
> refinement with the type traits for identity and inverse.
>
>>
>> Additionally, it is also possible to take a model of an AdditiveMagma 
>> and an operator that together are a model of a Magma. It might sound 
>> weird but I think this is kind of powerfull.
>>
> This was my intension to show that addition and multiplication are 
> only two instances of operation. Surely, the most important by far but 
> not the only ones.
>
> Peter
> ------------
> Peter Gottschling
> Research Associate
> Open Systems Laboratory
> Indiana University
> 301i Lindley Hall
> Bloomington, IN 47405
> Tel.: [phone]   Fax: [phone]
> http://www.osl.iu.edu/~pgottsch
>
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