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## [Axiom-developer] [#117 Inheritance of Monoid Structure in Direct Produc

 From: wyscc Subject: [Axiom-developer] [#117 Inheritance of Monoid Structure in Direct Product] Date: Fri, 04 Mar 2005 23:37:43 -0600

Changes
http://page.axiom-developer.org/zope/mathaction/117InheritanceOfMonoidStructureInDirectProduct/diff
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??changed:
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Does any one know why the direct product D of two (or more) copies of a monoid
R is not implemented as a monoid in Axiom?
The scalar multiplication is implemented coordinatewise, the identity element
is defined, but the monoid product between elements of D is not, and the domain
is not declared as a monoid. On the other hand, if R is a ring, then the direct
product is a ring. See <code>vector.spad</code>.

\begin{axiom}
NNI has Monoid
NNI2:= DirectProduct(2,NNI)
NNI2 has Monoid
a:NNI2:=directProduct([3,5])
3*a
b:NNI2:= 1
1*a
b*a
c:NNI2:=directProduct([1,1])
c*a
d:NNI2:=directProduct([1,2])
d*a
DirectProduct(2, INT) has Ring
\end{axiom}

Note how <I>smart</I> the Interpreter is to recognize that <code>c</code> is
really a scalar.

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