# XPolynomialsCat(vl, R)ΒΆ

The Category of polynomial rings with non-commutative variables. The coefficient ring may be non-commutative too. However coefficients commute with variables.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, R) -> %

from XFreeAlgebra(vl, R)

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

*: (vl, %) -> %

from XFreeAlgebra(vl, 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(vl, R)

coef: (%, FreeMonoid vl) -> R

from XFreeAlgebra(vl, R)

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

from CoercibleFrom FreeMonoid vl

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

from CoercibleFrom R

coerce: vl -> %

from XFreeAlgebra(vl, R)

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

from XFreeAlgebra(vl, R)

constant: % -> R

from XFreeAlgebra(vl, R)

degree: % -> NonNegativeInteger

`degree(p)` returns the degree of `p`. Note that the degree of a word is its length.

latex: % -> String

from SetCategory

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

lquo: (%, %) -> %

from XFreeAlgebra(vl, R)

lquo: (%, FreeMonoid vl) -> %

from XFreeAlgebra(vl, R)

lquo: (%, vl) -> %

from XFreeAlgebra(vl, R)

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

from XFreeAlgebra(vl, R)

maxdeg: % -> FreeMonoid vl

`maxdeg(p)` returns the greatest leading word in the support of `p`.

mindeg: % -> FreeMonoid vl

from XFreeAlgebra(vl, R)

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

from XFreeAlgebra(vl, R)

mirror: % -> %

from XFreeAlgebra(vl, R)

monomial?: % -> Boolean

from XFreeAlgebra(vl, R)

monomial: (R, FreeMonoid vl) -> %

from XFreeAlgebra(vl, R)

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

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

from XFreeAlgebra(vl, R)

quasiRegular: % -> %

from XFreeAlgebra(vl, R)

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: (%, %) -> %

from XFreeAlgebra(vl, R)

rquo: (%, FreeMonoid vl) -> %

from XFreeAlgebra(vl, R)

rquo: (%, vl) -> %

from XFreeAlgebra(vl, R)

sample: %

from AbelianMonoid

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

from XFreeAlgebra(vl, R)

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

from XFreeAlgebra(vl, R)

subtractIfCan: (%, %) -> Union(%, failed)
trunc: (%, NonNegativeInteger) -> %

`trunc(p, n)` returns the polynomial `p` truncated at order `n`.

varList: % -> List vl

from XFreeAlgebra(vl, R)

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

XFreeAlgebra(vl, R)