During the Aldor/Axiom Workshop 2008 at RISC in one of his presentations Stephen Watt began with:

  Q: When is a duck not a Duck?
  A: When it is an AbelianDuck!

This is in reference to the problem of defining appropriate categories for domains such as Integer which have a monoid consisting of (*,1) as well as a monoid consisting of (+,0) (or more specifically, a commutative group which is contained in such a monoid). The issue is: Why should we be required to use different names for categories like Monoid to mean just (*,1) and AbelianGroup to mean (+,0).

But an AbelianDuck really is a Duck!

The photographs at the bottom of this page where taken by me (Bill Page) during a walk to a small village called Friensdorf close to Hagenberg, Austria where the workshop was held. Which of these are Abelian? :-)

Here is another (still unsuccessful) attempt to express this in SPAD code. The error during compile:

  >> System error:
   |Identity;| [or a callee] requires more than one argument.

suggests an error in the compiler, but the specifications of the SPAD language are not clear on exactly whether this sort of construction involve passing symbols is really intended to work.

)abbrev category ASSOC Associative
++ m(m(a,b),c) = m(a,m(b,c))
Associative(m:Symbol):Category == with
  m:(%,%) -> %  ++ multiplication

)abbrev category COM Commutative
++ m(a,b) = m(b,a)
Commutative(m:Symbol):Category == with
       m:(%,%) -> %  ++ addition

)abbrev category ID Identity
++ m(u,a)=a and m(a,u)=a
Identity(u:Symbol,m:Symbol):Category == Commutative(m) with
       u: () -> %  ++ unit

)abbrev category INV Inverse
++ m(inv(a),a) = u and m(a,inv(a)) = u
Inverse(m:Symbol,inv:Symbol,u:Symbol):Category == Join(Commutative(m),Identity(u)) with
       inv: % -> %  ++ inverse

)abbrev category DIST Distributes
++ m(a,s(b,c)) = s(m(a,b),m(a,c))
++ m(s(a,b),c) = s(m(a,c),m(b,c))
Distributes(m,s):Category == with nil

)abbrev category MONOID Monoid
Monoid(m:Symbol,u:Symbol): Category
   == Join(Associative(m),Identity(u))

)abbrev category GROUP Group
Group(m:Symbol,inv:Symbol,u:Symbol): Category
   == Join(Monoid(m,u),Inverse(m,inv,u))

)abbrev category ABELG AbelianGroup
AbelianGroup(m:Symbol,inv:Symbol,u:Symbol): Category
   == Joint(Group(m,inv,u),Commutative(m))

)abbrev category RING Ring
Ring(s:Symbol,inv:Symbol,z:Symbol, m:Symbol,u:Symbol): Category
   == Join(AbelianGroup(s,inv,z),Monoid(m,u),Distributes(m,s))

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6486624088923745545-25px001.spad
      using old system compiler.
   ASSOC abbreviates category Associative 
------------------------------------------------------------------------
   initializing NRLIB ASSOC for Associative 
   compiling into NRLIB ASSOC 

;;;     ***       |Associative| REDEFINED
Time: 0 SEC.

   finalizing NRLIB ASSOC 
   Processing Associative for Browser database:
--------constructor---------
--------(m (% % %))---------
; compiling file "/var/aw/var/LatexWiki/ASSOC.NRLIB/ASSOC.lsp" (written 04 APR 2022 05:30:00 PM):

; /var/aw/var/LatexWiki/ASSOC.NRLIB/ASSOC.fasl written
; compilation finished in 0:00:00.004
------------------------------------------------------------------------
   Associative is now explicitly exposed in frame initial 
   Associative will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/ASSOC.NRLIB/ASSOC

   COM abbreviates category Commutative 
------------------------------------------------------------------------
   initializing NRLIB COM for Commutative 
   compiling into NRLIB COM 

;;;     ***       |Commutative| REDEFINED
Time: 0 SEC.

   finalizing NRLIB COM 
   Processing Commutative for Browser database:
--------constructor---------
--------(m (% % %))---------
; compiling file "/var/aw/var/LatexWiki/COM.NRLIB/COM.lsp" (written 04 APR 2022 05:30:00 PM):

; /var/aw/var/LatexWiki/COM.NRLIB/COM.fasl written
; compilation finished in 0:00:00.004
------------------------------------------------------------------------
   Commutative is now explicitly exposed in frame initial 
   Commutative will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/COM.NRLIB/COM

   ID abbreviates category Identity 
------------------------------------------------------------------------
   initializing NRLIB ID for Identity 
   compiling into NRLIB ID 

;;;     ***       |Identity| REDEFINED
Time: 0 SEC.

   finalizing NRLIB ID 
   Processing Identity for Browser database:
--------constructor---------
--------(u (%))---------
; compiling file "/var/aw/var/LatexWiki/ID.NRLIB/ID.lsp" (written 04 APR 2022 05:30:00 PM):

; /var/aw/var/LatexWiki/ID.NRLIB/ID.fasl written
; compilation finished in 0:00:00.004
------------------------------------------------------------------------

   >> System error:
   invalid number of arguments: 1

Except for the compile error, the following might work:

)sh Ring("+","-","0","*","1")

   The constructor Ring takes 0 arguments and you have given  5  .

)sh Ring(a,b,c,d,e)

   The constructor Ring takes 0 arguments and you have given  5  .

Then we could try this ...

)abbrev domain MYINT MyInteger
MyInteger: Ring(a,b,c,d,e) == add
  Rep:=Integer
  a(x: %, y: %): % == x
  b(x: %): % == x
  c(): % == 0 pretend %
  d(x: %, y: %): % == x
  e(): % == 0 pretend %

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4007632824524312372-25px004.spad
      using old system compiler.
   MYINT abbreviates domain MyInteger 
------------------------------------------------------------------------
   initializing NRLIB MYINT for MyInteger 
   compiling into NRLIB MYINT 

   >> System error:
   invalid number of arguments: 5