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

From: Andrey G. Grozin
Subject: Re: [Axiom-developer] Curiosities with Axiom mathematical structures
Date: Sun, 26 Feb 2006 19:53:54 +0600 (NOVT)

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.


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