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 vector.spad.

axiom
NNI has Monoid

(1)

axiom
NNI2:= DirectProduct(2,NNI)

(2)

axiom
NNI2 has Monoid

(3)

axiom
a:NNI2:=directProduct([3,5])

(4)

axiom
3*a

(5)

axiom
b:NNI2:= 1

(6)

axiom
1*a

(7)

axiom
b*a

(8)

axiom
c:NNI2:=directProduct([1,1])

(9)

axiom
c*a

(10)

axiom
d:NNI2:=directProduct([1,2])

(11)

axiom
d*a
   There are 34 exposed and 23 unexposed library operations named *
      having 2 argument(s) but none was determined to be applicable.
      Use HyperDoc Browse, or issue
                                )display op *
      to learn more about the available operations. Perhaps
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named *
      with argument type(s)
                     DirectProduct(2,NonNegativeInteger)
                     DirectProduct(2,NonNegativeInteger)
      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.
DirectProduct(2, INT) has Ring

(12)

Note how smart the Interpreter is to recognize that c is really a scalar.

Proposed patch in vector.spad

DirectProductCategory

DirectProduct

Bill, what is this property change ? What do you think if we change its status to fix proposed ?

I do not recall the change I made last year, but I agree to the status change

Status: open => fix proposed

Fixed in OpenAxiom

Fixed in FriCAS

Status: fix proposed => fixed somewhere