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Re: [glas] pure algebraic concepts cfr additive

From: Peter Gottschling (pgottsch_at_[hidden])
Date: 2005-05-26 19:02:28

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 Gottschling
Research Associate
Open Systems Laboratory
Indiana University
301i Lindley Hall
Bloomington, IN 47405
Tel.: +1 812 855-8898 Fax: +1 812 856 0853