Hello Willian,
thanks for the quick reply.
William Sit <address@hidden> writes:
 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.
I find that argument very unconvincing for several reasons:
1. "+" or "*" are *syntax*, not algebraic properties. Whether a
monoid is Abelian or not does stop it from being a monoid. The
mathematical definition of an Abelian monoid is that it is a
monoid, whose operation *additionally* is commutative.
2. The set of natural numbers, NN, has *many* Abelian structures on it,
for examples:
a. (NN, +, 0)  the usual addition
b. (NN, *, 1)  the usual multiplication, commutative!
c. (NN, max, 0)  "max" is the usual maximum function
Such a counterexample can be multiplied.