XPBWPolynomial(VarSet, R)

xlpoly.spad line 754 [edit on github]

• VarSet : OrderedSet
• R : CommutativeRing

This domain constructor implements polynomials in non-commutative variables written in the Poincare-Birkhoff-Witt basis from the Lyndon basis. These polynomials can be used to compute Baker-Campbell-Hausdorff relations. Author: Michel Petitot (petitot@lifl.fr).

* : (%, %) -> %

from Magma

* : (%, R) -> %

from RightModule(R)

* : (R, %) -> %

from LeftModule(R)

* : (VarSet, %) -> %

from XFreeAlgebra(VarSet, R)

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

LiePolyIfCan : % -> Union(LiePolynomial(VarSet, R), "failed")

LiePolyIfCan(p) return p if p is a Lie polynomial.

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associator : (%, %, %) -> %

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coef : (%, %) -> R

from XFreeAlgebra(VarSet, R)

coef : (%, FreeMonoid(VarSet)) -> R

from XFreeAlgebra(VarSet, R)

coefficient : (%, PoincareBirkhoffWittLyndonBasis(VarSet)) -> R

from FreeModuleCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

coefficients : % -> List(R)

from FreeModuleCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

coerce : R -> %

from XAlgebra(R)

coerce : VarSet -> %

from XFreeAlgebra(VarSet, R)

coerce : FreeMonoid(VarSet) -> %

from CoercibleFrom(FreeMonoid(VarSet))

coerce : Integer -> %

from NonAssociativeRing

coerce : LiePolynomial(VarSet, R) -> %

coerce(p) returns p.

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

coerce : % -> XDistributedPolynomial(VarSet, R)

coerce(p) returns p as a distributed polynomial.

coerce : % -> XRecursivePolynomial(VarSet, R)

coerce(p) returns p as a recursive polynomial.

commutator : (%, %) -> %

from NonAssociativeRng

constant : % -> R

from XFreeAlgebra(VarSet, R)

constant? : % -> Boolean

from XFreeAlgebra(VarSet, R)

construct : List(Record(k : PoincareBirkhoffWittLyndonBasis(VarSet), c : R)) -> %

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

constructOrdered : List(Record(k : PoincareBirkhoffWittLyndonBasis(VarSet), c : R)) -> %

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

degree : % -> NonNegativeInteger

from XPolynomialsCat(VarSet, R)

exp : (%, NonNegativeInteger) -> % if R has Module(Fraction(Integer))

exp(p, n) returns the exponential of p (truncated up to order n).

latex : % -> String

from SetCategory

leadingCoefficient : % -> R

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

leadingMonomial : % -> %

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

leadingSupport : % -> PoincareBirkhoffWittLyndonBasis(VarSet)

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

leadingTerm : % -> Record(k : PoincareBirkhoffWittLyndonBasis(VarSet), c : R)

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

linearExtend : (Mapping(R, PoincareBirkhoffWittLyndonBasis(VarSet)), %) -> R

from FreeModuleCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

listOfTerms : % -> List(Record(k : PoincareBirkhoffWittLyndonBasis(VarSet), c : R))

from IndexedDirectProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

log : (%, NonNegativeInteger) -> % if R has Module(Fraction(Integer))

log(p, n) returns the logarithm of p (truncated up to order n).

lquo : (%, %) -> %

from XFreeAlgebra(VarSet, R)

lquo : (%, VarSet) -> %

from XFreeAlgebra(VarSet, R)

lquo : (%, FreeMonoid(VarSet)) -> %

from XFreeAlgebra(VarSet, R)

map : (Mapping(R, R), %) -> %

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

maxdeg : % -> FreeMonoid(VarSet)

from XPolynomialsCat(VarSet, R)

mindeg : % -> FreeMonoid(VarSet)

from XFreeAlgebra(VarSet, R)

mindegTerm : % -> Record(k : FreeMonoid(VarSet), c : R)

from XFreeAlgebra(VarSet, R)

mirror : % -> %

from XFreeAlgebra(VarSet, R)

monomial : (R, FreeMonoid(VarSet)) -> %

from XFreeAlgebra(VarSet, R)

monomial : (R, PoincareBirkhoffWittLyndonBasis(VarSet)) -> %

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

monomial? : % -> Boolean

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

monomials : % -> List(%)

from FreeModuleCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

numberOfMonomials : % -> NonNegativeInteger

from IndexedDirectProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(R)

product : (%, %, NonNegativeInteger) -> %

product(a, b, n) returns a*b (truncated up to order n).

quasiRegular : % -> %

from XFreeAlgebra(VarSet, R)

quasiRegular? : % -> Boolean

from XFreeAlgebra(VarSet, R)

recip : % -> Union(%, "failed")

from MagmaWithUnit

reductum : % -> %

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

retract : % -> R

from RetractableTo(R)

retract : % -> FreeMonoid(VarSet)

from RetractableTo(FreeMonoid(VarSet))

retractIfCan : % -> Union(R, "failed")

from RetractableTo(R)

retractIfCan : % -> Union(FreeMonoid(VarSet), "failed")

from RetractableTo(FreeMonoid(VarSet))

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

rquo : (%, %) -> %

from XFreeAlgebra(VarSet, R)

rquo : (%, VarSet) -> %

from XFreeAlgebra(VarSet, R)

rquo : (%, FreeMonoid(VarSet)) -> %

from XFreeAlgebra(VarSet, R)

sample : () -> %

from AbelianMonoid

sh : (%, %) -> %

from XFreeAlgebra(VarSet, R)

sh : (%, NonNegativeInteger) -> %

from XFreeAlgebra(VarSet, R)

smaller? : (%, %) -> Boolean if R has Comparable

from Comparable

subtractIfCan : (%, %) -> Union(%, "failed")

from CancellationAbelianMonoid

support : % -> List(PoincareBirkhoffWittLyndonBasis(VarSet))

from FreeModuleCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

trunc : (%, NonNegativeInteger) -> %

from XPolynomialsCat(VarSet, R)

varList : % -> List(VarSet)

from XFreeAlgebra(VarSet, R)

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

Comparable

CoercibleFrom(R)

Algebra(R)

XAlgebra(R)

Monoid

AbelianMonoid

BiModule(R, R)

XFreeAlgebra(VarSet, R)

NonAssociativeAlgebra(R)

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

AbelianGroup

BiModule(%, %)

LeftModule(R)

noZeroDivisors

LeftModule(%)

RetractableTo(FreeMonoid(VarSet))

SetCategory

CoercibleFrom(FreeMonoid(VarSet))

Rng

IndexedDirectProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

Magma

XPolynomialsCat(VarSet, R)

SemiGroup

SemiRing

IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

CoercibleTo(OutputForm)

AbelianSemiGroup

NonAssociativeSemiRing

RightModule(%)

FreeModuleCategory(R, PoincareBirkhoffWittLyndonBasis(VarSet))

AbelianProductCategory(R)

Module(R)

RightModule(R)

RetractableTo(R)

NonAssociativeRng

unitsKnown

Ring

SemiRng

NonAssociativeSemiRng

BasicType