diff options
Diffstat (limited to 'src/algebra/xlpoly.spad.pamphlet')
| -rw-r--r-- | src/algebra/xlpoly.spad.pamphlet | 36 |
1 files changed, 6 insertions, 30 deletions
diff --git a/src/algebra/xlpoly.spad.pamphlet b/src/algebra/xlpoly.spad.pamphlet index 6090799a..0ce1f1b0 100644 --- a/src/algebra/xlpoly.spad.pamphlet +++ b/src/algebra/xlpoly.spad.pamphlet @@ -35,12 +35,9 @@ Magma(VarSet:OrderedSet):Public == Private where WORD ==> OrderedFreeMonoid(VarSet) EX ==> OutputForm - Public == Join(OrderedSet,RetractableTo VarSet) with + Public == Join(OrderedSet,RetractableTo VarSet,CoercibleTo WORD) with * : ($,$) -> $ ++ \axiom{x*y} returns the tree \axiom{[x,y]}. - coerce : $ -> WORD - ++ \axiom{coerce(x)} returns the element of \axiomType{OrderedFreeMonoid}(VarSet) - ++ corresponding to \axiom{x} by removing parentheses. first : $ -> VarSet ++ \axiom{first(x)} returns the first entry of the tree \axiom{x}. left : $ -> $ @@ -204,7 +201,7 @@ LyndonWord(VarSet:OrderedSet):Public == Private where OF ==> OutputForm ARRAY1==> OneDimensionalArray - Public == Join(OrderedSet,RetractableTo VarSet) with + Public == Join(OrderedSet,RetractableTo VarSet,CoercibleTo OFMON,CoercibleTo Magma VarSet) with retractable? : $ -> Boolean ++ \axiom{retractable?(x)} tests if \axiom{x} is a tree with only one entry. left : $ -> $ @@ -218,12 +215,6 @@ LyndonWord(VarSet:OrderedSet):Public == Private where lexico : ($,$) -> Boolean ++ \axiom{lexico(x,y)} returns \axiom{true} iff \axiom{x} is smaller than ++ \axiom{y} w.r.t. the lexicographical ordering induced by \axiom{VarSet}. - coerce : $ -> OFMON - ++ \axiom{coerce(x)} returns the element of \axiomType{OrderedFreeMonoid}(VarSet) - ++ corresponding to \axiom{x}. - coerce : $ -> Magma VarSet - ++ \axiom{coerce(x)} returns the element of \axiomType{Magma}(VarSet) - ++ corresponding to \axiom{x}. factor : OFMON -> List $ ++ \axiom{factor(x)} returns the decreasing factorization into Lyndon words. lyndon?: OFMON -> Boolean @@ -395,16 +386,12 @@ FreeLieAlgebra(VarSet:OrderedSet, R:CommutativeRing) :Category == CatDef where RN ==> Fraction Integer LWORD ==> LyndonWord(VarSet) - CatDef == Join(LieAlgebra(R)) with + CatDef == Join(LieAlgebra(R),CoercibleTo XDPOLY,CoercibleTo XRPOLY) with coef : (XRPOLY , $) -> R ++ \axiom{coef(x,y)} returns the scalar product of \axiom{x} by \axiom{y}, ++ the set of words being regarded as an orthogonal basis. coerce : VarSet -> $ ++ \axiom{coerce(x)} returns \axiom{x} as a Lie polynomial. - coerce : $ -> XDPOLY - ++ \axiom{coerce(x)} returns \axiom{x} as distributed polynomial. - coerce : $ -> XRPOLY - ++ \axiom{coerce(x)} returns \axiom{x} as a recursive polynomial. degree : $ -> NonNegativeInteger ++ \axiom{degree(x)} returns the greatest length of a word in the support of \axiom{x}. --if R has Module(RN) then @@ -743,12 +730,9 @@ PoincareBirkhoffWittLyndonBasis(VarSet: OrderedSet): Public == Private where NNI ==> NonNegativeInteger EX ==> OutputForm - Public == Join(OrderedSet, RetractableTo LWORD) with + Public == Join(OrderedSet, RetractableTo LWORD,CoercibleTo WORD) with 1: constant -> % ++ \spad{1} returns the empty list. - coerce : $ -> WORD - ++ \spad{coerce([l1]*[l2]*...[ln])} returns the word \spad{l1*l2*...*ln}, - ++ where \spad{[l_i]} is the backeted form of the Lyndon word \spad{l_i}. coerce : VarSet -> $ ++ \spad{coerce(v)} return \spad{v} first : $ -> LWORD @@ -866,13 +850,9 @@ XPBWPolynomial(VarSet:OrderedSet,R:CommutativeRing): XDPcat == XDPdef where I ==> Integer RN ==> Fraction(Integer) - XDPcat == Join(XPolynomialsCat(VarSet,R), FreeModuleCat(R, BASIS)) with + XDPcat == Join(XPolynomialsCat(VarSet,R), FreeModuleCat(R, BASIS),CoercibleTo XDPOLY,CoercibleTo XRPOLY) with coerce : LPOLY -> $ ++ \axiom{coerce(p)} returns \axiom{p}. - coerce : $ -> XDPOLY - ++ \axiom{coerce(p)} returns \axiom{p} as a distributed polynomial. - coerce : $ -> XRPOLY - ++ \axiom{coerce(p)} returns \axiom{p} as a recursive polynomial. LiePolyIfCan: $ -> Union(LPOLY,"failed") ++ \axiom{LiePolyIfCan(p)} return \axiom{p} if \axiom{p} is a Lie polynomial. product : ($,$,NNI) -> $ -- produit tronque a l'ordre n @@ -1111,17 +1091,13 @@ LieExponentials(VarSet, R, Order): XDPcat == XDPdef where TERM1 ==> Record(k:LWORD, c:R) EQ ==> Equation(R) - XDPcat == Group with + XDPcat == Join(Group,CoercibleTo XDPOLY,CoercibleTo PBWPOLY) with exp : LPOLY -> $ ++ \axiom{exp(p)} returns the exponential of \axiom{p}. log : $ -> LPOLY ++ \axiom{log(p)} returns the logarithm of \axiom{p}. ListOfTerms : $ -> LTERMS ++ \axiom{ListOfTerms(p)} returns the internal representation of \axiom{p}. - coerce : $ -> XDPOLY - ++ \axiom{coerce(g)} returns the internal representation of \axiom{g}. - coerce : $ -> PBWPOLY - ++ \axiom{coerce(g)} returns the internal representation of \axiom{g}. mirror : $ -> $ ++ \axiom{mirror(g)} is the mirror of the internal representation of \axiom{g}. varList : $ -> List VarSet |