XPolynomialsCat(vl, R)

xpoly.spad line 108 [edit on github]

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

* : (%, %) -> %
from Magma
* : (%, R) -> %
from XFreeAlgebra(vl, R)
* : (R, %) -> %
from LeftModule(R)
* : (vl, %) -> %
from XFreeAlgebra(vl, 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
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
annihilate? : (%, %) -> Boolean
from Rng
antiCommutator : (%, %) -> %
from NonAssociativeSemiRng
associator : (%, %, %) -> %
from NonAssociativeRng
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
coef : (%, %) -> R
from XFreeAlgebra(vl, R)
coef : (%, FreeMonoid(vl)) -> R
from XFreeAlgebra(vl, R)
coerce : R -> %
from XAlgebra(R)
coerce : vl -> %
from XFreeAlgebra(vl, R)
coerce : FreeMonoid(vl) -> %
from CoercibleFrom(FreeMonoid(vl))
coerce : Integer -> %
from NonAssociativeRing
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from NonAssociativeRng
constant : % -> R
from XFreeAlgebra(vl, R)
constant? : % -> Boolean
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 : (%, vl) -> %
from XFreeAlgebra(vl, R)
lquo : (%, FreeMonoid(vl)) -> %
from XFreeAlgebra(vl, R)
map : (Mapping(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 : (R, FreeMonoid(vl)) -> %
from XFreeAlgebra(vl, R)
monomial? : % -> Boolean
from XFreeAlgebra(vl, R)
one? : % -> Boolean
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
plenaryPower : (%, PositiveInteger) -> % if R has CommutativeRing
from NonAssociativeAlgebra(R)
quasiRegular : % -> %
from XFreeAlgebra(vl, R)
quasiRegular? : % -> Boolean
from XFreeAlgebra(vl, R)
recip : % -> Union(%, "failed")
from MagmaWithUnit
retract : % -> R
from RetractableTo(R)
retract : % -> FreeMonoid(vl)
from RetractableTo(FreeMonoid(vl))
retractIfCan : % -> Union(R, "failed")
from RetractableTo(R)
retractIfCan : % -> Union(FreeMonoid(vl), "failed")
from RetractableTo(FreeMonoid(vl))
rightPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower : (%, PositiveInteger) -> %
from Magma
rightRecip : % -> Union(%, "failed")
from MagmaWithUnit
rquo : (%, %) -> %
from XFreeAlgebra(vl, R)
rquo : (%, vl) -> %
from XFreeAlgebra(vl, R)
rquo : (%, FreeMonoid(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")
from CancellationAbelianMonoid
trunc : (%, NonNegativeInteger) -> %

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

varList : % -> List(vl)
from XFreeAlgebra(vl, R)
zero? : % -> Boolean
from AbelianMonoid
~= : (%, %) -> Boolean
from BasicType

CoercibleFrom(R)

Algebra(R)

noZeroDivisors

RightModule(%)

Monoid

BiModule(R, R)

NonAssociativeAlgebra(R)

CoercibleFrom(FreeMonoid(vl))

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

AbelianGroup

NonAssociativeSemiRng

RetractableTo(FreeMonoid(vl))

LeftModule(R)

LeftModule(%)

SetCategory

BiModule(%, %)

Rng

SemiGroup

Magma

XAlgebra(R)

unitsKnown

CoercibleTo(OutputForm)

AbelianSemiGroup

NonAssociativeSemiRing

Module(R)

XFreeAlgebra(vl, R)

RightModule(R)

RetractableTo(R)

NonAssociativeRng

Ring

SemiRng

AbelianMonoid

BasicType

SemiRing