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

 From: William Sit Subject: Re: [Axiom-developer] Curiosities with Axiom mathematical structures Date: Sun, 26 Feb 2006 09:10:59 -0500

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).
>
>   Does anyone know the reason of those curiosities?

One probable reason may be the convention in mathematics to use + instead of *
when a semigroup, monoid, or group is abelian. Another is the independence
allows for the category of rings without having to overload the * operation for
both the underlying abelian group and the multiplicative monoid, which would
have been very confusing! A third one is because addition is an ubiquitous
operation in many algebraic objects (vector space, modules, rings, algebras,
etc.) with richer structures.

The second is the most compelling. It can be generalized to a principle for
setting up new basic categories: every time a set is equipped with a new
(usually binary) operation with some axioms to satisfy, a new category that
encapsulates the operation with a new notation is necessary in order to allow
for simultaneous operations on the same set, providing a rich algebraic
structure. An example would be Lie algebras which have at least an addition, a
multiplication, and the Lie bracket. Square matrices of a fixed size form a Lie
algebra.

William

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