# XPBWPolynomial(VarSet, R)ΒΆ

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`).

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, R) -> %

from RightModule R

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

*: (VarSet, %) -> %

from XFreeAlgebra(VarSet, R)

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
associator: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
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: % -> 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.

coerce: FreeMonoid VarSet -> %

from CoercibleFrom FreeMonoid VarSet

coerce: Integer -> %
coerce: LiePolynomial(VarSet, R) -> %

`coerce(p)` returns `p`.

coerce: R -> %

from CoercibleFrom R

coerce: VarSet -> %

from XFreeAlgebra(VarSet, R)

commutator: (%, %) -> %
constant?: % -> Boolean

from XFreeAlgebra(VarSet, R)

constant: % -> R

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

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis VarSet)

from IndexedProductCategory(R, 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

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

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

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

from FreeModuleCategory(R, PoincareBirkhoffWittLyndonBasis VarSet)

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

from XFreeAlgebra(VarSet, R)

lquo: (%, VarSet) -> %

from XFreeAlgebra(VarSet, R)

map: (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?: % -> Boolean

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis VarSet)

monomial: (R, FreeMonoid VarSet) -> %

from XFreeAlgebra(VarSet, R)

monomial: (R, PoincareBirkhoffWittLyndonBasis VarSet) -> %

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis VarSet)

monomials: % -> List %

from FreeModuleCategory(R, PoincareBirkhoffWittLyndonBasis VarSet)

numberOfMonomials: % -> NonNegativeInteger
one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

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

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

quasiRegular?: % -> Boolean

from XFreeAlgebra(VarSet, R)

quasiRegular: % -> %

from XFreeAlgebra(VarSet, R)

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(R, PoincareBirkhoffWittLyndonBasis VarSet)

retract: % -> FreeMonoid VarSet

from RetractableTo FreeMonoid VarSet

retract: % -> R

from RetractableTo R

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

from RetractableTo FreeMonoid VarSet

retractIfCan: % -> Union(R, failed)

from RetractableTo R

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

rquo: (%, %) -> %

from XFreeAlgebra(VarSet, R)

rquo: (%, FreeMonoid VarSet) -> %

from XFreeAlgebra(VarSet, R)

rquo: (%, 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)
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

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

CoercibleFrom FreeMonoid VarSet

Comparable if R has Comparable

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has noZeroDivisors

RetractableTo FreeMonoid VarSet

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown

XFreeAlgebra(VarSet, R)

XPolynomialsCat(VarSet, R)