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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.

spad
)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))
spad
   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 31 JUL 2013 06:48:07 PM):
; /var/aw/var/LatexWiki/ASSOC.NRLIB/ASSOC.fasl written ; compilation finished in 0:00:00.005 ------------------------------------------------------------------------ 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 (% % %))--------- --------constructor--------- --------(m (% % %))--------- --->-->Commutative(): Spurious comments: \spad{m}(a,{}\spad{b}) = \spad{m}(\spad{b},{}a) ; compiling file "/var/aw/var/LatexWiki/COM.NRLIB/COM.lsp" (written 31 JUL 2013 06:48:07 PM):
; /var/aw/var/LatexWiki/COM.NRLIB/COM.fasl written ; compilation finished in 0:00:00.002 ------------------------------------------------------------------------ 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--------- --------(m (% % %))--------- --------constructor--------- --------(m (% % %))--------- --------constructor--------- --------(u (%))--------- --->-->Identity(): Spurious comments: \spad{m}(a,{}\spad{b}) = \spad{m}(\spad{b},{}a) --->-->Identity(): Spurious comments: \spad{m}(\spad{u},{}a)=a and \spad{m}(a,{}\spad{u})=a ; compiling file "/var/aw/var/LatexWiki/ID.NRLIB/ID.lsp" (written 31 JUL 2013 06:48:07 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:

fricas
)sh Ring("+","-","0","*","1")
The constructor Ring takes 0 arguments and you have given 5 . Ring("+","-","0","*","1") has commutative("+")
The constructor Ring takes 0 arguments and you have given 5 .

fricas
)sh Ring(a,b,c,d,e)
The constructor Ring takes 0 arguments and you have given 5 . Ring(a,b,c,d,e) has commutative(a)
The constructor Ring takes 0 arguments and you have given 5 .

Then we could try this ...

spad
)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 %
spad
   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




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