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last edited 15 years ago by Bill Page |
Edit detail for SandBoxAbelianDuck revision 4 of 8
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Editor: Bill Page
Time: 2008/07/27 14:20:06 GMT-7 |
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| Note: ducks in a row | ||
changed: --- m(m(a,b),c) = m(a,m(b,c)) ++ m(m(a,b),c) = m(a,m(b,c)) changed: - m:(%,%) -> % m:(%,%) -> % ++ multiplication changed: --- m(a,b) = m(b,a) ++ m(a,b) = m(b,a) changed: - m:(%,%) -> % m:(%,%) -> % ++ addition changed: --- m(u,a)=a and m(a,u)=a ++ m(u,a)=a and m(a,u)=a changed: - u: () -> % u: () -> % ++ unit changed: --- m(inv(a),a) = u and m(a,inv(a)) = u ++ m(inv(a),a) = u and m(a,inv(a)) = u changed: - inv: % -> % inv: % -> % ++ inverse changed: --- m(a,s(b,c)) = s(m(a,b),m(a,c)) --- m(s(a,b),c) = s(m(a,c),m(b,c)) ++ m(a,s(b,c)) = s(m(a,b),m(a,c)) ++ m(s(a,b),c) = s(m(a,c),m(b,c))
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!
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/zope2/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: 0.01 SEC.
finalizing NRLIB ASSOC
Processing Associative for Browser database:
--------(m (% % %))---------
--------constructor---------
------------------------------------------------------------------------
Associative is now explicitly exposed in frame initial
Associative will be automatically loaded when needed from
/var/zope2/var/LatexWiki/ASSOC.NRLIB/code
COM abbreviates category Commutative
------------------------------------------------------------------------
initializing NRLIB COM for Commutative
compiling into NRLIB COM
;;; *** |Commutative| REDEFINED
Time: 0.01 SEC.
finalizing NRLIB COM
Processing Commutative for Browser database:
--------(m (% % %))---------
--------constructor---------
------------------------------------------------------------------------
Commutative is now explicitly exposed in frame initial
Commutative will be automatically loaded when needed from
/var/zope2/var/LatexWiki/COM.NRLIB/code
ID abbreviates category Identity
------------------------------------------------------------------------
initializing NRLIB ID for Identity
compiling into NRLIB ID
;;; *** |Identity| REDEFINED
Time: 0.01 SEC.
finalizing NRLIB ID
Processing Identity for Browser database:
--------(u (%))---------
--------constructor---------
------------------------------------------------------------------------
>> System error:
|Identity;| [or a callee] requires more than one argument.axiom)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 .
Let's try this ...
axiom)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 .
Interesting! Is it somewhere written that "has" can have a category as its first argument?
OK, we are going to implement...
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/zope2/var/LatexWiki/4007632824524312372-25px004.spad using
old system compiler.
MYINT abbreviates domain MyInteger
------------------------------------------------------------------------
initializing NRLIB MYINT for MyInteger
compiling into NRLIB MYINT
>> System error:
EVAL [or a callee] requires less than five arguments.