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On Fri, May 15, 2009 at 03:23:33AM +0200, Franz Lehner wrote:

Now it more or less works, included is a first example of a bialgebra (just too lazy to write up the antipode):

The line below is no longer needed, since the constructors are included in FriCAS

)library TENSORC TENSORP TENSORD

spad
)abbrev category COALG CoAlgebra
CoAlgebra(R : CommutativeRing, RR : TensorProductCategory(R, %, %)) : Category == _
        Module(R) with
    delta : % -> RR
       ++ delta is comultiplication
    counit: % -> R
       ++ counit
)abbrev category BIALG BiAlgebra
BiAlgebra(R : CommutativeRing, RR : TensorProductCategory(R, %, %)) : Category == _
     Join(Algebra(R), CoAlgebra(R, RR))
)abbrev category HOPFALG HopfAlgebra
HopfAlgebra(R : CommutativeRing, RR : TensorProductCategory(R, %, %)) : Category _
     == BiAlgebra(R, RR) with
    S : % -> %
       ++ the antipode
)abbrev domain PHALG PolyHopfAlgebra
PolyHopfAlgebra(R: CommutativeRing,x: Symbol): C == T where
    S == Variable x
    FM == FreeMonoid Symbol
    PxP == TensorProduct(R, FM, FM, %, %) 
--    C == Join(FreeModule(R, FM),HopfAlgebra(R, PxP))
    C == Join(UnivariatePolynomialCategory(R), _
         FreeModuleCategory(R, FM), BiAlgebra(R, PxP))
TERM == Record(k: FM,c: R)
T == SparseUnivariatePolynomial(R) add Rep:= SparseUnivariatePolynomial(R)
monom(a:FM,r:R):% == one? a => monomial(r, 0@NonNegativeInteger) monomial(r, nthExpon(a, 1@Integer))
-- monomial(r:R, n:NonNegativeInteger) == monomial(r,n)@Rep
coerce(p:%):OutputForm == outputForm(p,outputForm x)$Rep
delta1(n:NonNegativeInteger):PxP == res: PxP := 0 nn:List NonNegativeInteger := [k for k in 0..n] for k1 in nn for k2 in reverse nn repeat res:= res + binomial(n,k1)*tensor(monomial(1@R,k1),monomial(1@R,k2))$PxP res
listOfTerms(p:%):List TERM == res:List TERM := [] while not zero? p repeat m:FM := (variable()$S)^degree p m1:TERM := [m,leadingCoefficient p] res:=concat!(res, m1) p:=reductum p res
delta(p:%):PxP == zero? p => 0 lt:R := leadingCoefficient p lt * delta1(degree p) + delta reductum p
counit(p:%):R == coefficient(p,0)
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1189790658020391427-25px001.spad
      using old system compiler.
   COALG abbreviates category CoAlgebra 
------------------------------------------------------------------------
   initializing NRLIB COALG for CoAlgebra 
   compiling into NRLIB COALG 
;;; *** |CoAlgebra| REDEFINED Time: 0 SEC.
finalizing NRLIB COALG Processing CoAlgebra for Browser database: --->-->CoAlgebra(constructor): Not documented!!!! --------(delta (RR %))--------- --------(counit (R %))--------- --->-->CoAlgebra(): Missing Description ; compiling file "/var/aw/var/LatexWiki/COALG.NRLIB/COALG.lsp" (written 31 JUL 2013 03:46:39 PM):
; /var/aw/var/LatexWiki/COALG.NRLIB/COALG.fasl written ; compilation finished in 0:00:00.006 ------------------------------------------------------------------------ CoAlgebra is now explicitly exposed in frame initial CoAlgebra will be automatically loaded when needed from /var/aw/var/LatexWiki/COALG.NRLIB/COALG
BIALG abbreviates category BiAlgebra ------------------------------------------------------------------------ initializing NRLIB BIALG for BiAlgebra compiling into NRLIB BIALG
;;; *** |BiAlgebra| REDEFINED Time: 0 SEC.
finalizing NRLIB BIALG Processing BiAlgebra for Browser database: --->-->BiAlgebra(constructor): Not documented!!!! --------(delta (RR %))--------- --------(counit (R %))--------- --->-->BiAlgebra(): Missing Description ; compiling file "/var/aw/var/LatexWiki/BIALG.NRLIB/BIALG.lsp" (written 31 JUL 2013 03:46:39 PM):
; /var/aw/var/LatexWiki/BIALG.NRLIB/BIALG.fasl written ; compilation finished in 0:00:00.003 ------------------------------------------------------------------------ BiAlgebra is now explicitly exposed in frame initial BiAlgebra will be automatically loaded when needed from /var/aw/var/LatexWiki/BIALG.NRLIB/BIALG
HOPFALG abbreviates category HopfAlgebra ------------------------------------------------------------------------ initializing NRLIB HOPFALG for HopfAlgebra compiling into NRLIB HOPFALG
;;; *** |HopfAlgebra| REDEFINED Time: 0 SEC.
finalizing NRLIB HOPFALG Processing HopfAlgebra for Browser database: --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/HOPFALG.spad-->HopfAlgebra(constructor): Not documented!!!! --------(delta (RR %))--------- --------(counit (R %))--------- --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/HOPFALG.spad-->HopfAlgebra(constructor): Not documented!!!! --------(S (% %))--------- --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/HOPFALG.spad-->HopfAlgebra(): Missing Description ; compiling file "/var/aw/var/LatexWiki/HOPFALG.NRLIB/HOPFALG.lsp" (written 31 JUL 2013 03:46:39 PM):
; /var/aw/var/LatexWiki/HOPFALG.NRLIB/HOPFALG.fasl written ; compilation finished in 0:00:00.004 ------------------------------------------------------------------------ HopfAlgebra is now explicitly exposed in frame initial HopfAlgebra will be automatically loaded when needed from /var/aw/var/LatexWiki/HOPFALG.NRLIB/HOPFALG
PHALG abbreviates domain PolyHopfAlgebra ------------------------------------------------------------------------ initializing NRLIB PHALG for PolyHopfAlgebra compiling into NRLIB PHALG compiling exported monom : (FreeMonoid Symbol,R) -> $ Time: 0.05 SEC.
compiling exported coerce : $ -> OutputForm Time: 0 SEC.
compiling local delta1 : NonNegativeInteger -> TensorProduct(R,FreeMonoid Symbol,FreeMonoid Symbol,$,$) Time: 0.02 SEC.
compiling exported listOfTerms : $ -> List Record(k: FreeMonoid Symbol,c: R) Time: 0.01 SEC.
compiling exported delta : $ -> TensorProduct(R,FreeMonoid Symbol,FreeMonoid Symbol,$,$) Time: 0.01 SEC.
compiling exported counit : $ -> R Time: 0 SEC.
****** Domain: $ already in scope augmenting $: (CharacteristicNonZero) ****** Domain: R already in scope augmenting R: (PolynomialFactorizationExplicit) ****** Domain: R already in scope augmenting R: (ConvertibleTo (InputForm)) ****** Domain: (SingletonAsOrderedSet) already in scope augmenting (SingletonAsOrderedSet): (ConvertibleTo (InputForm)) ****** Domain: R already in scope augmenting R: (ConvertibleTo (Pattern (Float))) ****** Domain: (SingletonAsOrderedSet) already in scope augmenting (SingletonAsOrderedSet): (ConvertibleTo (Pattern (Float))) ****** Domain: R already in scope augmenting R: (ConvertibleTo (Pattern (Integer))) ****** Domain: (SingletonAsOrderedSet) already in scope augmenting (SingletonAsOrderedSet): (ConvertibleTo (Pattern (Integer))) ****** Domain: R already in scope augmenting R: (OrderedAbelianMonoidSup) ****** Domain: R already in scope augmenting R: (PatternMatchable (Float)) ****** Domain: (SingletonAsOrderedSet) already in scope augmenting (SingletonAsOrderedSet): (PatternMatchable (Float)) ****** Domain: R already in scope augmenting R: (PatternMatchable (Integer)) ****** Domain: (SingletonAsOrderedSet) already in scope augmenting (SingletonAsOrderedSet): (PatternMatchable (Integer)) ****** Domain: R already in scope augmenting R: (Algebra (Fraction (Integer))) ****** Domain: R already in scope augmenting R: (EntireRing) ****** Domain: R already in scope augmenting R: (Field) ****** Domain: R already in scope augmenting R: (GcdDomain) ****** Domain: R already in scope augmenting R: (IntegralDomain) ****** Domain: R already in scope augmenting R: (LinearlyExplicitRingOver (Integer)) ****** Domain: R already in scope augmenting R: (PartialDifferentialRing (Symbol)) ****** Domain: R already in scope augmenting R: (PolynomialFactorizationExplicit) ****** Domain: R already in scope augmenting R: (RetractableTo (Fraction (Integer))) ****** Domain: R already in scope augmenting R: (RetractableTo (Integer)) ****** Domain: R already in scope augmenting R: (StepThrough) (time taken in buildFunctor: 260)
;;; *** |PolyHopfAlgebra| REDEFINED
;;; *** |PolyHopfAlgebra| REDEFINED Time: 0.27 SEC.
Warnings: [1] not known that (Comparable) is of mode (CATEGORY domain (SIGNATURE * ($ (Symbol) $)) (SIGNATURE * ($ $ (Symbol))) (SIGNATURE ^ ($ (Symbol) (NonNegativeInteger))) (SIGNATURE hclf ($ $ $)) (SIGNATURE hcrf ($ $ $)) (SIGNATURE lquo ((Union $ failed) $ $)) (SIGNATURE rquo ((Union $ failed) $ $)) (SIGNATURE divide ((Union (Record (: lm $) (: rm $)) failed) $ $)) (SIGNATURE overlap ((Record (: lm $) (: mm $) (: rm $)) $ $)) (SIGNATURE size ((NonNegativeInteger) $)) (SIGNATURE factors ((List (Record (: gen (Symbol)) (: exp (NonNegativeInteger)))) $)) (SIGNATURE nthExpon ((NonNegativeInteger) $ (Integer))) (SIGNATURE nthFactor ((Symbol) $ (Integer))) (SIGNATURE mapExpon ($ (Mapping (NonNegativeInteger) (NonNegativeInteger)) $)) (SIGNATURE mapGen ($ (Mapping (Symbol) (Symbol)) $)) (IF (has (Symbol) (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch)) [2] not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE * ($ (Symbol) $)) (SIGNATURE * ($ $ (Symbol))) (SIGNATURE ^ ($ (Symbol) (NonNegativeInteger))) (SIGNATURE hclf ($ $ $)) (SIGNATURE hcrf ($ $ $)) (SIGNATURE lquo ((Union $ failed) $ $)) (SIGNATURE rquo ((Union $ failed) $ $)) (SIGNATURE divide ((Union (Record (: lm $) (: rm $)) failed) $ $)) (SIGNATURE overlap ((Record (: lm $) (: mm $) (: rm $)) $ $)) (SIGNATURE size ((NonNegativeInteger) $)) (SIGNATURE factors ((List (Record (: gen (Symbol)) (: exp (NonNegativeInteger)))) $)) (SIGNATURE nthExpon ((NonNegativeInteger) $ (Integer))) (SIGNATURE nthFactor ((Symbol) $ (Integer))) (SIGNATURE mapExpon ($ (Mapping (NonNegativeInteger) (NonNegativeInteger)) $)) (SIGNATURE mapGen ($ (Mapping (Symbol) (Symbol)) $)) (IF (has (Symbol) (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch)) [3] delta1: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE * ($ (Symbol) $)) (SIGNATURE * ($ $ (Symbol))) (SIGNATURE ^ ($ (Symbol) (NonNegativeInteger))) (SIGNATURE hclf ($ $ $)) (SIGNATURE hcrf ($ $ $)) (SIGNATURE lquo ((Union $ failed) $ $)) (SIGNATURE rquo ((Union $ failed) $ $)) (SIGNATURE divide ((Union (Record (: lm $) (: rm $)) failed) $ $)) (SIGNATURE overlap ((Record (: lm $) (: mm $) (: rm $)) $ $)) (SIGNATURE size ((NonNegativeInteger) $)) (SIGNATURE factors ((List (Record (: gen (Symbol)) (: exp (NonNegativeInteger)))) $)) (SIGNATURE nthExpon ((NonNegativeInteger) $ (Integer))) (SIGNATURE nthFactor ((Symbol) $ (Integer))) (SIGNATURE mapExpon ($ (Mapping (NonNegativeInteger) (NonNegativeInteger)) $)) (SIGNATURE mapGen ($ (Mapping (Symbol) (Symbol)) $)) (IF (has (Symbol) (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch))
Cumulative Statistics for Constructor PolyHopfAlgebra Time: 0.36 seconds
--------------non extending category---------------------- .. PolyHopfAlgebra(#1,#2) of cat (|Join| (|UnivariatePolynomialCategory| |#1|) (|FreeModuleCategory| |#1| (|FreeMonoid| (|Symbol|))) (|BiAlgebra| |#1| (|TensorProduct| |#1| (|FreeMonoid| (|Symbol|)) (|FreeMonoid| (|Symbol|)) $ $))) has no outputForm : (%,OutputForm) -> OutputForm finalizing NRLIB PHALG Processing PolyHopfAlgebra for Browser database: --->-->PolyHopfAlgebra(constructor): Not documented!!!! --------(delta (RR %))--------- --------(counit (R %))--------- --->-->PolyHopfAlgebra(constructor): Not documented!!!! --------(S (% %))--------- --->-->PolyHopfAlgebra(): Missing Description ; compiling file "/var/aw/var/LatexWiki/PHALG.NRLIB/PHALG.lsp" (written 31 JUL 2013 03:46:39 PM):
; /var/aw/var/LatexWiki/PHALG.NRLIB/PHALG.fasl written ; compilation finished in 0:00:00.043 ------------------------------------------------------------------------ PolyHopfAlgebra is now explicitly exposed in frame initial PolyHopfAlgebra will be automatically loaded when needed from /var/aw/var/LatexWiki/PHALG.NRLIB/PHALG

fricas
P:=PolyHopfAlgebra(Integer,'u)

\label{eq1}\hbox{\axiomType{PolyHopfAlgebra}\ } (\hbox{\axiomType{Integer}\ } , u)(1)
Type: Type
fricas
p:=monomial(1,1)$P

\label{eq2}u(2)
Type: PolyHopfAlgebra?(Integer,u)
fricas
delta p

\label{eq3}{u \otimes 1}+{1 \otimes u}(3)
Type: TensorProduct(Integer,FreeMonoid?(Symbol),FreeMonoid?(Symbol),PolyHopfAlgebra?(Integer,u),PolyHopfAlgebra?(Integer,u))
fricas
delta(p^2)

\label{eq4}{{{u}^{2}}\otimes 1}+{2 \ {u \otimes u}}+{1 \otimes{{u}^{2}}}(4)
Type: TensorProduct(Integer,FreeMonoid?(Symbol),FreeMonoid?(Symbol),PolyHopfAlgebra?(Integer,u),PolyHopfAlgebra?(Integer,u))
fricas
delta(p)^2
Function: monomial : (Integer,FreeMonoid(Symbol)) -> % is missing from domain: PolyHopfAlgebra(Integer,u) Internal Error The function monomial with signature $(Integer)(FreeMonoid (Symbol)) is missing from domain PolyHopfAlgebra(Integer)u

Please comment.

Next comes the old problem come with iterated structures. Is it in principle possible to have:

  TensorProduct(R:CommutativeRing,LB:List OrderedSet, LM:List FreeModuleCategory(R))?

Since the coproduct is coassociative, we might want to define "powers", but I realize that tensor powers are actually feasible.

Actually the basis is not redundant; in the attached example the implementation of the Coalgebra does not use the basis which it pretends to the tensor product. Still exporting a basis would be a nice thing.

More questions:

  1. How far are we from having unicode in output?
  2. CoAlgebra? or Coalgebra?
  3. delta or coproduct? epsilon or counit? S or antipode? or both?
  4. I do not see which domains should use TensorProductProperty?, the target module S is not part of the domain. Anyways morphisms have to be implemented in packages ("TensorProductFunctions2?" or something like that)
  5. For now TensorProduct relies on FreeModuleCategory. To make use of the existing applicable domains we have to add FreeModuleCategory everywhere. Extend would be good ... First of course comes a cleanup of FreeModule itself. The example I cooked up in the attachment is rather ugly and fragile.
  6. The compiler does not warn about missing exports. (this is a really nice feature of the aldor compiler) I can add all categories I like, they pass the compiler and )sh domain shows all the operations exported by the categories. The only way to detect missing ones is by runtime errors.
  7. ceterum censeo Aldorem esse liberandam (ok, that suffices for now)




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