# XFreeAlgebra(vl, R)ΒΆ

This category specifies operations for polynomials and formal series with non-commutative variables.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, R) -> %

`x * r` returns the product of `x` by `r`. Usefull if `R` is a non-commutative Ring.

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

*: (vl, %) -> %

`v * x` returns the product of a variable `x` by `x`.

+: (%, %) -> %

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

`coef(x, y)` returns scalar product of `x` by `y`, the set of words being regarded as an orthogonal basis.

coef: (%, FreeMonoid vl) -> R

`coef(x, w)` returns the coefficient of the word `w` in `x`.

coerce: % -> OutputForm
coerce: FreeMonoid vl -> %

from CoercibleFrom FreeMonoid vl

coerce: Integer -> %
coerce: R -> %

from CoercibleFrom R

coerce: vl -> %

`coerce(v)` returns `v`.

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

`constant?(x)` returns `true` if `x` is constant.

constant: % -> R

`constant(x)` returns the constant term of `x`.

latex: % -> String

from SetCategory

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

lquo: (%, %) -> %

`lquo(x, y)` returns the left simplification of `x` by `y`.

lquo: (%, FreeMonoid vl) -> %

`lquo(x, w)` returns the left simplification of `x` by the word `w`.

lquo: (%, vl) -> %

`lquo(x, v)` returns the left simplification of `x` by the variable `v`.

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

`map(fn, x)` returns `Sum(fn(r_i) w_i)` if `x` writes `Sum(r_i w_i)`.

mindeg: % -> FreeMonoid vl

`mindeg(x)` returns the little word which appears in `x`. Error if `x=0`.

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

`mindegTerm(x)` returns the term whose word is `mindeg(x)`.

mirror: % -> %

`mirror(x)` returns `Sum(r_i mirror(w_i))` if `x` writes `Sum(r_i w_i)`.

monomial?: % -> Boolean

`monomial?(x)` returns `true` if `x` is a monomial

monomial: (R, FreeMonoid vl) -> %

`monomial(r, w)` returns the product of the word `w` by the coefficient `r`.

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> % if R has CommutativeRing
quasiRegular?: % -> Boolean

`quasiRegular?(x)` return `true` if `constant(x)` is zero.

quasiRegular: % -> %

`quasiRegular(x)` return `x` minus its constant term.

recip: % -> Union(%, failed)

from MagmaWithUnit

retract: % -> FreeMonoid vl

from RetractableTo FreeMonoid vl

retract: % -> R

from RetractableTo R

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

from RetractableTo FreeMonoid vl

retractIfCan: % -> Union(R, failed)

from RetractableTo R

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

rquo: (%, %) -> %

`rquo(x, y)` returns the right simplification of `x` by `y`.

rquo: (%, FreeMonoid vl) -> %

`rquo(x, w)` returns the right simplification of `x` by `w`.

rquo: (%, vl) -> %

`rquo(x, v)` returns the right simplification of `x` by the variable `v`.

sample: %

from AbelianMonoid

sh: (%, %) -> % if R has CommutativeRing

`sh(x, y)` returns the shuffle-product of `x` by `y`. This multiplication is associative and commutative.

sh: (%, NonNegativeInteger) -> % if R has CommutativeRing

`sh(x, n)` returns the shuffle power of `x` to the `n`.

subtractIfCan: (%, %) -> Union(%, failed)
varList: % -> List vl

`varList(x)` returns the list of variables which appear in `x`.

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

Magma

MagmaWithUnit

Module R if R has CommutativeRing

Monoid

NonAssociativeAlgebra R if R has CommutativeRing

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has noZeroDivisors

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown