Re: [Axiom-developer] Curiosities with Axiom mathematical structures

[Andrey G. Grozin]

On Sun, 26 Feb 2006, Gabriel Dos Reis wrote:
>  In the impressive diagram titled "Basic Agebra Hierarchy" displayed
> in the Axiom Book (I only have a copy of the edition copyrighted 1992,
> NAG), AbelianSemiGroup is not "derived" from SemiGroup, and similarly
> AbelianMonoid is not "derived" from Monoid.  I find that curious as it
> goes counter the mathematical fact that an AbelianMonoid *is* a
> Monoid, with an additional algebraic law (commutation).
As far as I know, the reason is simple and purely notational. SemiGroup 
has the binary operation *, and AbelianSemiGroup has +. Therefore, it 
cannot be derived from SemiGroup. But some other categories can be derived 
from both of them, inheriting * (maybe non-commutative) from SemiGroup and 
+ (always commutative) from AbelianSemiGroup.

My copy of the Axiom book is in my office, and I am writing from home, so, 
details may be wrong.

Andrey