Martin's beautiful idea:

We can use the Aldor extend construct to add MyMonoid(...) as a category to a previously defined domain.

(1) -> <aldor>
#include "fricas"

MyMonoid(T: Type, m: (T, T) -> T): Category == with {
   square:T -> T;
   coerce:T -> OutputForm;
   default {
     -- This hack for output only works for domains that
     -- have a representation known to Lisp.
     coerce(x:T):OutputForm == x pretend OutputForm;
     square(t: T): T == m(t,t)
   }
}</aldor>

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/mymonoid.as 
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.

Compiling Lisp source code from file ./mymonoid.lsp
   Issuing )library command for mymonoid

Reading /var/aw/var/LatexWiki/mymonoid.asy
   MyMonoid is now explicitly exposed in frame initial 
   MyMonoid will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/mymonoid

#include "fricas"
#library MyMonoid "mymonoid.ao";
import from MyMonoid;
MyWord: with { 
   coerce: String -> %;
   c:(%, %) -> %;
   d:(%, %) -> %;
}
   == add {
   Rep == String;
   import from String;
   coerce(a: String): % == per(a);
   c(a: %, b: %):%  == per(concat(rep(a), rep(b)));
   d(a: %, b: %):%  == a;
}

import from MyWord;
extend MyWord: MyMonoid(MyWord, c) == add;

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7928879980347790310-25px002.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
   Compiling Lisp source code from file 
      ./7928879980347790310-25px002.lsp
   Issuing )library command for 7928879980347790310-25px002
   Reading /var/aw/var/LatexWiki/7928879980347790310-25px002.asy
   MyWord is now explicitly exposed in frame initial 
   MyWord will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/7928879980347790310-25px002

This is what Axiom sees:

)sh MyMonoid

 MyMonoid(T: Type,m: ((T, T) -> T)) is a category constructor
 Abbreviation for MyMonoid is MYMONOI 
 This constructor is exposed in this frame.
------------------------------- Operations --------------------------------

)sh MyWord

 MyWord is a domain constructor.
 Abbreviation for MyWord is MYWORD 
 This constructor is exposed in this frame.
 4 Names for 5 Operations in this Domain.
------------------------------- Operations --------------------------------

 c : (%, %) -> %                       coerce : MyWord -> OutputForm
 coerce : String -> %                  d : (%, %) -> %
 square : MyWord -> MyWord

Try it:

a := "Bingo"::MyWord

(1)

square a

(2)

MyWord has MyMonoid(MyWord, c)

(3)

That's pretty cool that FriCAS knows MyWord is a MyMonoid!

But:

MyWord has MyMonoid(MyWord, d)

(4)

Oops, that's not cool! :( Aldor and the FriCAS interpreter ought to be able to do better than this.

Here are some more examples:

#include "fricas"
#library MyMonoid "mymonoid.ao";
import from MyMonoid;
MyInt: with { 
   coerce: Integer -> %;
   +:(%, %) -> %;
}
   == add {
   Rep == Integer;
   import from Integer;
   coerce(a: Integer): % == per(a);
   (a:%) + (b:%):%  == per(rep(a)+rep(b))
}
import from MyInt;
extend MyInt: MyMonoid(MyInt, +) == add;

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/774031628094989668-25px006.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
   Compiling Lisp source code from file 
      ./774031628094989668-25px006.lsp
   Issuing )library command for 774031628094989668-25px006
   Reading /var/aw/var/LatexWiki/774031628094989668-25px006.asy
   MyInt is now explicitly exposed in frame initial 
   MyInt will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/774031628094989668-25px006

This is what FriCAS sees:

)sh MyInt

 MyInt is a domain constructor.
 Abbreviation for MyInt is MYINT 
 This constructor is exposed in this frame.
 3 Names for 4 Operations in this Domain.
------------------------------- Operations --------------------------------

 ?+? : (%, %) -> %                     coerce : MyInt -> OutputForm
 coerce : Integer -> %                 square : MyInt -> MyInt

Try it:

b := 3::MyInt

(5)

b+1

(6)

square b

(7)

This is very general. Notice that we can rename the monoid operation from * to +!

#include "fricas"
#library MyMonoid "mymonoid.ao";
import from MyMonoid;
MyFloat: with { 
   coerce: DoubleFloat -> %;
   +:(%, %) -> %;
}
   == add {
   Rep == DoubleFloat;
   import from DoubleFloat;
   coerce(a: DoubleFloat): % == per(a);
   (a:%) + (b:%):%  == per(rep(a)*rep(b))
}
import from MyFloat;
extend MyFloat: MyMonoid(MyFloat, +) == add;

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5335661638819928165-25px009.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
   Compiling Lisp source code from file 
      ./5335661638819928165-25px009.lsp
   Issuing )library command for 5335661638819928165-25px009
   Reading /var/aw/var/LatexWiki/5335661638819928165-25px009.asy
   MyFloat is now explicitly exposed in frame initial 
   MyFloat will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/5335661638819928165-25px009

This is what FriCAS sees:

)sh MyFloat

 MyFloat is a domain constructor.
 Abbreviation for MyFloat is MYFLOAT 
 This constructor is exposed in this frame.
 3 Names for 4 Operations in this Domain.
------------------------------- Operations --------------------------------

 ?+? : (%, %) -> %                     coerce : MyFloat -> OutputForm
 coerce : DoubleFloat -> %             square : MyFloat -> MyFloat

Try it:

f := 3.1::DoubleFloat::MyFloat

   >> Error detected within library code:
   Unrecognized atom in OutputForm

Here's a variation on the example dirprod.as posted by Ralf Hemmecke on Mon, 13 Mar 2006 14:33:34 +0100 '[Axiom-developer]? Re: BINGO,Curiosities with Axiom mathematical structures'

#include "fricas"
define DirProdCat(n: Integer, R: Type): Category == with {
    identity: % -> %;
}
DirProd(n: Integer, R: Type): DirProdCat(n, R) == add {
    Rep == List Integer; -- dummy implementation
    import from Rep;
    identity(x: %): % == x;
 }

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/mydirprod.as 
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
   Compiling Lisp source code from file ./mydirprod.lsp
   Issuing )library command for mydirprod
   Reading /var/aw/var/LatexWiki/mydirprod.asy
   DirProdCat is now explicitly exposed in frame initial 
   DirProdCat will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/mydirprod
   DirProd is now explicitly exposed in frame initial 
   DirProd will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/mydirprod

Try it:

x:= Integer;

y:= NonNegativeInteger;

DPx ==> DirProd(2, x);

DPx has DirProdCat(2, x)

(8)

DPx has DirProdCat(2, y)

(9)

DPx has DirProdCat(3, x)

(10)

DPx has DirProdCat(3, y)

(11)

The results show that the Aldor compiler treats a domain (R) and an element of a domain (n) differently in terms of information retained for the has operation. Had the Type in R: Type been replaced by Symbol and the lines defining x, y removed, all results would have been true. Notice here, contrary to Ralf's example, both the DirProdCat and DirProd constructors totally ignored the parameters. The two constructions DirProd(2,x) and DirProd(2,y) are really identical in implemetation. So the only information to distinguish DirProdCat(2,x) and DirProdCat(2,y) must have come from the declaration of the parameters (on the left of :).

)show DirProd(2,x)

 DirProd(2,Integer) is a domain constructor.
 Abbreviation for DirProd is DirProd 
 This constructor is exposed in this frame.
 1 Names for 1 Operations in this Domain.
------------------------------- Operations --------------------------------

 identity : % -> %                    

)show DirProd(2,y)

 DirProd(2,NonNegativeInteger) is a domain constructor.
 Abbreviation for DirProd is DirProd 
 This constructor is exposed in this frame.
 1 Names for 1 Operations in this Domain.
------------------------------- Operations --------------------------------

 identity : % -> %