
From:  Andrey G. Grozin 
Subject:  Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures 
Date:  Sun, 26 Feb 2006 19:53:54 +0600 (NOVT) 
On Sun, 26 Feb 2006, Gabriel Dos Reis wrote:
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 noncommutative) from SemiGroup and + (always commutative) from AbelianSemiGroup.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).
My copy of the Axiom book is in my office, and I am writing from home, so, details may be wrong.
Andrey
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